Nelson Functions 11
1st Edition
ISBN: 9780176332037
Textbook solutions
All Solutions
Section 4-2: Working with Integer Exponents
Exercise 1
Step 1
1 of 2
[begin{gathered}
{text{In this excercise we shall rewrite each expression}}{text{.}} hfill \
{text{a) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{5^{ – 4}} = frac{1}{{{5^4}}} hfill \
hfill \
{text{b) Remember this rule, }}{left( {frac{a}{b}} right)^{ – n}} = frac{1}{{{{left( {frac{a}{b}} right)}^n}}} = {left( {frac{b}{a}} right)^n} hfill \
{left( { – frac{1}{{10}}} right)^{ – 3}} = frac{1}{{{{left( { – frac{1}{{10}}} right)}^3}}} hfill \
hfill \
{text{Therefore,}} hfill \
{left( { – frac{1}{{10}}} right)^{ – 3}} = {left( { – 10} right)^3} hfill \
hfill \
{text{c) Remember this rule, }}{left( {frac{a}{b}} right)^{ – n}} = frac{1}{{{{left( {frac{a}{b}} right)}^n}}} = {left( {frac{b}{a}} right)^n} hfill \
frac{1}{{{2^{ – 4}}}} = left( {frac{1}{{frac{1}{{{2^4}}}}}} right) hfill \
{text{Therefore,}} hfill \
frac{1}{{{2^{ – 4}}}} = {2^4} hfill \
hfill \
{text{d) Remember this rule, }}{left( {frac{a}{b}} right)^{ – n}} = frac{1}{{{{left( {frac{a}{b}} right)}^n}}} = {left( {frac{b}{a}} right)^n} hfill \
{left( {frac{3}{{11}}} right)^{ – 1}} = frac{1}{{{{left( {frac{3}{{11}}} right)}^1}}} hfill \
{text{Therefore,}} hfill \
{left( {frac{3}{{11}}} right)^{ – 1}} = frac{{11}}{3} hfill \
hfill \
{text{e) Remember this rule, }}{left( {frac{a}{b}} right)^{ – n}} = frac{1}{{{{left( {frac{a}{b}} right)}^n}}} = {left( {frac{b}{a}} right)^n} hfill \
frac{{{7^{ – 2}}}}{{{8^{ – 1}}}} = frac{1}{{left( {frac{{{7^{ – 2}}}}{{{8^{ – 1}}}}} right)}} hfill \
{text{Therefore,}} hfill \
frac{{{7^{ – 2}}}}{{{8^{ – 1}}}} = frac{{{8^{ – 1}}}}{{{7^{ – 2}}}} hfill \
hfill \
end{gathered} ]
{text{In this excercise we shall rewrite each expression}}{text{.}} hfill \
{text{a) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{5^{ – 4}} = frac{1}{{{5^4}}} hfill \
hfill \
{text{b) Remember this rule, }}{left( {frac{a}{b}} right)^{ – n}} = frac{1}{{{{left( {frac{a}{b}} right)}^n}}} = {left( {frac{b}{a}} right)^n} hfill \
{left( { – frac{1}{{10}}} right)^{ – 3}} = frac{1}{{{{left( { – frac{1}{{10}}} right)}^3}}} hfill \
hfill \
{text{Therefore,}} hfill \
{left( { – frac{1}{{10}}} right)^{ – 3}} = {left( { – 10} right)^3} hfill \
hfill \
{text{c) Remember this rule, }}{left( {frac{a}{b}} right)^{ – n}} = frac{1}{{{{left( {frac{a}{b}} right)}^n}}} = {left( {frac{b}{a}} right)^n} hfill \
frac{1}{{{2^{ – 4}}}} = left( {frac{1}{{frac{1}{{{2^4}}}}}} right) hfill \
{text{Therefore,}} hfill \
frac{1}{{{2^{ – 4}}}} = {2^4} hfill \
hfill \
{text{d) Remember this rule, }}{left( {frac{a}{b}} right)^{ – n}} = frac{1}{{{{left( {frac{a}{b}} right)}^n}}} = {left( {frac{b}{a}} right)^n} hfill \
{left( {frac{3}{{11}}} right)^{ – 1}} = frac{1}{{{{left( {frac{3}{{11}}} right)}^1}}} hfill \
{text{Therefore,}} hfill \
{left( {frac{3}{{11}}} right)^{ – 1}} = frac{{11}}{3} hfill \
hfill \
{text{e) Remember this rule, }}{left( {frac{a}{b}} right)^{ – n}} = frac{1}{{{{left( {frac{a}{b}} right)}^n}}} = {left( {frac{b}{a}} right)^n} hfill \
frac{{{7^{ – 2}}}}{{{8^{ – 1}}}} = frac{1}{{left( {frac{{{7^{ – 2}}}}{{{8^{ – 1}}}}} right)}} hfill \
{text{Therefore,}} hfill \
frac{{{7^{ – 2}}}}{{{8^{ – 1}}}} = frac{{{8^{ – 1}}}}{{{7^{ – 2}}}} hfill \
hfill \
end{gathered} ]
Result
2 of 2
a) $dfrac{1}{5^4}$
b) $(-10)^3$
c) $2^4$
d) $-left(dfrac{5}{6}right)^3$
e) $dfrac{11}{3}$
f) $dfrac{8}{7^2}$
Exercise 2
Step 1
1 of 2
[begin{gathered}
hfill \
hfill \
{text{a) Remember this rule, }}{a^n}left( {{a^m}} right) = {a^{n + m}} hfill \
{left( { – 10} right)^8}{left( { – 10} right)^{ – 8}} = – {10^{8 + left( { – 8} right)}} hfill \
{left( { – 10} right)^8}{left( { – 10} right)^{ – 8}} = – {10^0} hfill \
{text{Therefore,}} hfill \
{left( { – 10} right)^8}{left( { – 10} right)^{ – 8}} = 1 hfill \
hfill \
{text{b) Remember this rule, }}{a^n}left( {{a^m}} right) = {a^{n + m}} hfill \
{6^{ – 7}}left( {{6^5}} right) = {6^{ – 7 + 5}} hfill \
{6^{ – 7}}left( {{6^5}} right) = {6^{ – 2}} hfill \
{text{Therefore,}} hfill \
{6^{ – 7}}left( {{6^5}} right) = frac{1}{{{6^2}}} hfill \
hfill \
{text{c) Remember this rule, }}{{text{a}}^n} div {a^m} = {a^{n – m}} hfill \
frac{{{2^8}}}{{{2^{ – 5}}}} = {2^{8 – left( { – 5} right)}} hfill \
frac{{{2^8}}}{{{2^{ – 5}}}} = {2^{13}} hfill \
{text{Therefore,}} hfill \
frac{{{2^8}}}{{{2^{ – 5}}}} = frac{1}{{{2^{13}}}} hfill \
hfill \
{text{d) Remember this rule, }}{{text{a}}^n} div {a^m} = {a^{n – m}} hfill \
frac{{{{11}^{ – 3}}}}{{{{11}^5}}} = {11^{ – 3 – 5}} hfill \
frac{{{{11}^{ – 3}}}}{{{{11}^5}}} = {11^{ – 8}} hfill \
{text{Therefore,}} hfill \
frac{{{{11}^{ – 3}}}}{{{{11}^5}}} = frac{1}{{{{11}^8}}} hfill \
hfill \
{text{e) Remember this rule, (}}{a^n}{)^m} = {a^{n cdot m}} hfill \
{left( { – {9^4}} right)^{ – 1}} = – {9^{4left( { – 1} right)}} hfill \
{left( { – {9^4}} right)^{ – 1}} = – {9^{ – 4}} hfill \
{text{Therefore,}} hfill \
{left( { – {9^4}} right)^{ – 1}} = – frac{1}{{{9^4}}} hfill \
hfill \
{text{f) Remember this rule, (}}{a^n}{)^m} = {a^{n cdot m}} hfill \
{left[ {{{left( {{7^{ – 3}}} right)}^{ – 2}}} right]^{ – 2}} = left( {{7^{( – 3)left( { – 2} right)( – 2)}}} right) hfill \
{left[ {{{left( {{7^{ – 3}}} right)}^{ – 2}}} right]^{ – 2}} = {7^{ – 12}} hfill \
{text{Therefore,}} hfill \
{left[ {{{left( {{7^{ – 3}}} right)}^{ – 2}}} right]^{ – 2}} = frac{1}{{{7^{12}}}} hfill \
hfill \
hfill \
end{gathered} ]
hfill \
hfill \
{text{a) Remember this rule, }}{a^n}left( {{a^m}} right) = {a^{n + m}} hfill \
{left( { – 10} right)^8}{left( { – 10} right)^{ – 8}} = – {10^{8 + left( { – 8} right)}} hfill \
{left( { – 10} right)^8}{left( { – 10} right)^{ – 8}} = – {10^0} hfill \
{text{Therefore,}} hfill \
{left( { – 10} right)^8}{left( { – 10} right)^{ – 8}} = 1 hfill \
hfill \
{text{b) Remember this rule, }}{a^n}left( {{a^m}} right) = {a^{n + m}} hfill \
{6^{ – 7}}left( {{6^5}} right) = {6^{ – 7 + 5}} hfill \
{6^{ – 7}}left( {{6^5}} right) = {6^{ – 2}} hfill \
{text{Therefore,}} hfill \
{6^{ – 7}}left( {{6^5}} right) = frac{1}{{{6^2}}} hfill \
hfill \
{text{c) Remember this rule, }}{{text{a}}^n} div {a^m} = {a^{n – m}} hfill \
frac{{{2^8}}}{{{2^{ – 5}}}} = {2^{8 – left( { – 5} right)}} hfill \
frac{{{2^8}}}{{{2^{ – 5}}}} = {2^{13}} hfill \
{text{Therefore,}} hfill \
frac{{{2^8}}}{{{2^{ – 5}}}} = frac{1}{{{2^{13}}}} hfill \
hfill \
{text{d) Remember this rule, }}{{text{a}}^n} div {a^m} = {a^{n – m}} hfill \
frac{{{{11}^{ – 3}}}}{{{{11}^5}}} = {11^{ – 3 – 5}} hfill \
frac{{{{11}^{ – 3}}}}{{{{11}^5}}} = {11^{ – 8}} hfill \
{text{Therefore,}} hfill \
frac{{{{11}^{ – 3}}}}{{{{11}^5}}} = frac{1}{{{{11}^8}}} hfill \
hfill \
{text{e) Remember this rule, (}}{a^n}{)^m} = {a^{n cdot m}} hfill \
{left( { – {9^4}} right)^{ – 1}} = – {9^{4left( { – 1} right)}} hfill \
{left( { – {9^4}} right)^{ – 1}} = – {9^{ – 4}} hfill \
{text{Therefore,}} hfill \
{left( { – {9^4}} right)^{ – 1}} = – frac{1}{{{9^4}}} hfill \
hfill \
{text{f) Remember this rule, (}}{a^n}{)^m} = {a^{n cdot m}} hfill \
{left[ {{{left( {{7^{ – 3}}} right)}^{ – 2}}} right]^{ – 2}} = left( {{7^{( – 3)left( { – 2} right)( – 2)}}} right) hfill \
{left[ {{{left( {{7^{ – 3}}} right)}^{ – 2}}} right]^{ – 2}} = {7^{ – 12}} hfill \
{text{Therefore,}} hfill \
{left[ {{{left( {{7^{ – 3}}} right)}^{ – 2}}} right]^{ – 2}} = frac{1}{{{7^{12}}}} hfill \
hfill \
hfill \
end{gathered} ]
Result
2 of 2
a) $1$
b) $dfrac{1}{6^2}$
c) $2^{13}$
d) $dfrac{1}{11^8}$
e) $-dfrac{1}{9^4}$
f) $dfrac{1}{7^{12}}$
Exercise 3
Step 1
1 of 2
[begin{gathered}
{text{In this exercise we shall determine which is the greater power}} hfill \
{text{a) }}{2^{ – 5}} = frac{1}{{{2^5}}} = 0.03125{text{ }} hfill \
{text{ }}{left( {frac{1}{2}} right)^{ – 5}} = {2^5} = 32 hfill \
{text{Therefore, }}{left( {frac{1}{2}} right)^{ – 5}}{text{ is greater since }}32 > 0.03125 hfill \
end{gathered} ]
{text{In this exercise we shall determine which is the greater power}} hfill \
{text{a) }}{2^{ – 5}} = frac{1}{{{2^5}}} = 0.03125{text{ }} hfill \
{text{ }}{left( {frac{1}{2}} right)^{ – 5}} = {2^5} = 32 hfill \
{text{Therefore, }}{left( {frac{1}{2}} right)^{ – 5}}{text{ is greater since }}32 > 0.03125 hfill \
end{gathered} ]
Result
2 of 2
$2^{-5}$ is less than $left(dfrac{1}{2}right)^{-5}$
Exercise 4
Step 1
1 of 7
a) $2^{-3}(2^7)=2^{-3+7}=2^4=16$
Use the following rule:
$$
x^acdot x^b=x^{a+b}
$$
Step 2
2 of 7
b) $(-8)^3(-8)^{-3}=(-8)^{3+(-3)}=(-8)^0=1$
Use the following rule:
$x^acdot x^b=x^{a+b}$
$x^0=1$ ; $xneq 0$
Step 3
3 of 7
c) $dfrac{5^4}{5^6}=5^{4-6}=5^{-2}=dfrac{1}{5^2}=dfrac{1}{25}$
Use the following rule:
$dfrac{x^a}{x^b}=x^{a-b}$
$$
x^{-a}=dfrac{1}{x^a}
$$
Step 4
4 of 7
d) $dfrac{3^{-8}}{3^{-6}}=3^{-8-(-6)}=3^{-2}=dfrac{1}{3^2}=dfrac{1}{9}$
Use the following rule:
$dfrac{x^a}{x^b}=x^{a-b}$
$$
x^{-a}=dfrac{1}{x^a}
$$
Step 5
5 of 7
e) $(4^{-3})^{-1}=4^3=64$
Use the following rule:
$$
(x^a)^b=x^{ab}
$$
Step 6
6 of 7
f) $(7^{-1})^2=7^{-2}=dfrac{1}{7^2}=dfrac{1}{49}$
Use the following rule:
$(x^a)^b=x^{ab}$
$$
x^{-a}=dfrac{1}{x^a}
$$
Result
7 of 7
a) $2^4=16$
b) $(-8)^0=1$
c) $5^{-2}=dfrac{1}{25}$
d) $3^{-2}=dfrac{1}{9}$
e) $4^3=64$
f) $7^{-2}=dfrac{1}{49}$
Exercise 6
Step 1
1 of 7
[begin{gathered}
{text{In this excercise we shall evaluate and simplify the following}} hfill \
{text{a)Remember this rule, }}{a^n}left( {{a^m}} right) = {a^{n + m}}{text{ }} hfill \
left( {10} right)left( {{{10}^4}left( {{{10}^{ – 2}}} right)} right) = left( {10} right){10^{4 + left( { – 2} right)}} hfill \
left( {10} right)left( {{{10}^4}left( {{{10}^{ – 2}}} right)} right) = left( {10} right){10^2} hfill \
{text{Remember this rule, }}{a^n}left( {{a^m}} right) = {a^{n + m}}{text{ }} hfill \
left( {10} right)left( {{{10}^4}left( {{{10}^{ – 2}}} right)} right) = {10^{1 + 2}} hfill \
left( {10} right)left( {{{10}^4}left( {{{10}^{ – 2}}} right)} right) = {10^2} hfill \
left( {10} right)left( {{{10}^4}left( {{{10}^{ – 2}}} right)} right) = 1000 hfill \
end{gathered} ]
{text{In this excercise we shall evaluate and simplify the following}} hfill \
{text{a)Remember this rule, }}{a^n}left( {{a^m}} right) = {a^{n + m}}{text{ }} hfill \
left( {10} right)left( {{{10}^4}left( {{{10}^{ – 2}}} right)} right) = left( {10} right){10^{4 + left( { – 2} right)}} hfill \
left( {10} right)left( {{{10}^4}left( {{{10}^{ – 2}}} right)} right) = left( {10} right){10^2} hfill \
{text{Remember this rule, }}{a^n}left( {{a^m}} right) = {a^{n + m}}{text{ }} hfill \
left( {10} right)left( {{{10}^4}left( {{{10}^{ – 2}}} right)} right) = {10^{1 + 2}} hfill \
left( {10} right)left( {{{10}^4}left( {{{10}^{ – 2}}} right)} right) = {10^2} hfill \
left( {10} right)left( {{{10}^4}left( {{{10}^{ – 2}}} right)} right) = 1000 hfill \
end{gathered} ]
Step 2
2 of 7
[begin{gathered}
{text{b) Remember this rule, }}{a^n}left( {{a^m}} right) = {a^{n + m}}{text{ }} hfill \
8left( {{8^2}} right)left( {{8^{ – 4}}} right) = {8^{1 + 2 + left( { – 4} right)}} hfill \
8left( {{8^2}} right)left( {{8^{ – 4}}} right) = {8^{ – 1}} hfill \
{text{Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
8left( {{8^2}} right)left( {{8^{ – 4}}} right) = frac{1}{{{8^{ – 1}}}} hfill \
8left( {{8^2}} right)left( {{8^{ – 4}}} right) = frac{1}{8} hfill \
end{gathered} ]
{text{b) Remember this rule, }}{a^n}left( {{a^m}} right) = {a^{n + m}}{text{ }} hfill \
8left( {{8^2}} right)left( {{8^{ – 4}}} right) = {8^{1 + 2 + left( { – 4} right)}} hfill \
8left( {{8^2}} right)left( {{8^{ – 4}}} right) = {8^{ – 1}} hfill \
{text{Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
8left( {{8^2}} right)left( {{8^{ – 4}}} right) = frac{1}{{{8^{ – 1}}}} hfill \
8left( {{8^2}} right)left( {{8^{ – 4}}} right) = frac{1}{8} hfill \
end{gathered} ]
Step 3
3 of 7
[begin{gathered}
{text{c) Simplify the denominator}}{text{.}} hfill \
frac{{left( {{6^{ – 5}}} right)}}{{{{left( {{6^2}} right)}^{ – 2}}}} = frac{{left( {{6^{ – 5}}} right)}}{{{6^{2left( { – 2} right)}}}} hfill \
frac{{left( {{6^{ – 5}}} right)}}{{{{left( {{6^2}} right)}^{ – 2}}}} = frac{{{6^{ – 5}}}}{{{6^{ – 4}}}} hfill \
{text{Remember this rule, }}{{text{a}}^n} div {a^m} = {a^{n – m}} hfill \
frac{{left( {{6^{ – 5}}} right)}}{{{{left( {{6^2}} right)}^{ – 2}}}} = {6^{ – 5 – 4}} hfill \
frac{{left( {{6^{ – 5}}} right)}}{{{{left( {{6^2}} right)}^{ – 2}}}} = {6^{ – 1}} hfill \
{text{Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
frac{{left( {{6^{ – 5}}} right)}}{{{{left( {{6^2}} right)}^{ – 2}}}} = frac{1}{{{6^{ – 1}}}} hfill \
frac{{left( {{6^{ – 5}}} right)}}{{{{left( {{6^2}} right)}^{ – 2}}}} = frac{1}{6} hfill \
end{gathered} ]
{text{c) Simplify the denominator}}{text{.}} hfill \
frac{{left( {{6^{ – 5}}} right)}}{{{{left( {{6^2}} right)}^{ – 2}}}} = frac{{left( {{6^{ – 5}}} right)}}{{{6^{2left( { – 2} right)}}}} hfill \
frac{{left( {{6^{ – 5}}} right)}}{{{{left( {{6^2}} right)}^{ – 2}}}} = frac{{{6^{ – 5}}}}{{{6^{ – 4}}}} hfill \
{text{Remember this rule, }}{{text{a}}^n} div {a^m} = {a^{n – m}} hfill \
frac{{left( {{6^{ – 5}}} right)}}{{{{left( {{6^2}} right)}^{ – 2}}}} = {6^{ – 5 – 4}} hfill \
frac{{left( {{6^{ – 5}}} right)}}{{{{left( {{6^2}} right)}^{ – 2}}}} = {6^{ – 1}} hfill \
{text{Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
frac{{left( {{6^{ – 5}}} right)}}{{{{left( {{6^2}} right)}^{ – 2}}}} = frac{1}{{{6^{ – 1}}}} hfill \
frac{{left( {{6^{ – 5}}} right)}}{{{{left( {{6^2}} right)}^{ – 2}}}} = frac{1}{6} hfill \
end{gathered} ]
Step 4
4 of 7
[begin{gathered}
{text{d) Simplify the denominator}}{text{.}} hfill \
frac{{{4^{ – 10}}}}{{{{({4^{ – 4}})}^3}}} = frac{{{4^{ – 10}}}}{{{4^{ – 4left( 3 right)}}}} hfill \
frac{{{4^{ – 10}}}}{{{{({4^{ – 4}})}^3}}} = frac{{{4^{ – 10}}}}{{{4^{12}}}} hfill \
{text{Remember this rule, }}{{text{a}}^n} div {a^m} = {a^{n – m}} hfill \
frac{{{4^{ – 10}}}}{{{{({4^{ – 4}})}^3}}} = {4^{ – 10 – (12)}} hfill \
frac{{{4^{ – 10}}}}{{{{({4^{ – 4}})}^3}}} = {4^2} hfill \
frac{{{4^{ – 10}}}}{{{{({4^{ – 4}})}^3}}} = 16 hfill \
end{gathered} ]
{text{d) Simplify the denominator}}{text{.}} hfill \
frac{{{4^{ – 10}}}}{{{{({4^{ – 4}})}^3}}} = frac{{{4^{ – 10}}}}{{{4^{ – 4left( 3 right)}}}} hfill \
frac{{{4^{ – 10}}}}{{{{({4^{ – 4}})}^3}}} = frac{{{4^{ – 10}}}}{{{4^{12}}}} hfill \
{text{Remember this rule, }}{{text{a}}^n} div {a^m} = {a^{n – m}} hfill \
frac{{{4^{ – 10}}}}{{{{({4^{ – 4}})}^3}}} = {4^{ – 10 – (12)}} hfill \
frac{{{4^{ – 10}}}}{{{{({4^{ – 4}})}^3}}} = {4^2} hfill \
frac{{{4^{ – 10}}}}{{{{({4^{ – 4}})}^3}}} = 16 hfill \
end{gathered} ]
Step 5
5 of 7
[begin{gathered}
{text{e)}} hfill \
{{text{2}}^8}left( {frac{{{2^{ – 5}}}}{{{2^6}}}} right) = {2^8}left( {{2^{ – 5 – (6)}}} right) hfill \
{{text{2}}^8}left( {frac{{{2^{ – 5}}}}{{{2^6}}}} right) = {2^8}left( {{2^{ – 11}}} right) hfill \
{{text{2}}^8}left( {frac{{{2^{ – 5}}}}{{{2^6}}}} right) = {2^{8 + left( { – 11} right)}} hfill \
{{text{2}}^8}left( {frac{{{2^{ – 5}}}}{{{2^6}}}} right) = {2^{ – 3}} hfill \
{{text{2}}^8}left( {frac{{{2^{ – 5}}}}{{{2^6}}}} right) = frac{1}{{{2^{ – 3}}}} hfill \
{{text{2}}^8}left( {frac{{{2^{ – 5}}}}{{{2^6}}}} right) = frac{1}{8} hfill \
end{gathered} ]
{text{e)}} hfill \
{{text{2}}^8}left( {frac{{{2^{ – 5}}}}{{{2^6}}}} right) = {2^8}left( {{2^{ – 5 – (6)}}} right) hfill \
{{text{2}}^8}left( {frac{{{2^{ – 5}}}}{{{2^6}}}} right) = {2^8}left( {{2^{ – 11}}} right) hfill \
{{text{2}}^8}left( {frac{{{2^{ – 5}}}}{{{2^6}}}} right) = {2^{8 + left( { – 11} right)}} hfill \
{{text{2}}^8}left( {frac{{{2^{ – 5}}}}{{{2^6}}}} right) = {2^{ – 3}} hfill \
{{text{2}}^8}left( {frac{{{2^{ – 5}}}}{{{2^6}}}} right) = frac{1}{{{2^{ – 3}}}} hfill \
{{text{2}}^8}left( {frac{{{2^{ – 5}}}}{{{2^6}}}} right) = frac{1}{8} hfill \
end{gathered} ]
Step 6
6 of 7
[begin{gathered}
{text{f) Simplify the expression}}{text{.}} hfill \
{text{1}}{{text{3}}^{ – 5}}{left( {frac{{{{13}^2}}}{{{{13}^8}}}} right)^{ – 1}} = {13^{ – 5}}{left( {{{13}^{2 – 8}}} right)^{ – 1}} hfill \
{text{1}}{{text{3}}^{ – 5}}{left( {frac{{{{13}^2}}}{{{{13}^8}}}} right)^{ – 1}} = {13^{ – 5}}{left( {{{13}^{ – 6}}} right)^{ – 1}} hfill \
{text{Remember this rule, (}}{a^n}{)^m} = {a^{n cdot m}} hfill \
{text{1}}{{text{3}}^{ – 5}}{left( {frac{{{{13}^2}}}{{{{13}^8}}}} right)^{ – 1}} = {13^{ – 5}}left( {{{13}^{ – 6left( { – 1} right)}}} right) hfill \
{text{Remember this rule, }}{a^n}left( {{a^m}} right) = {a^{n + m}}{text{ }} hfill \
{text{1}}{{text{3}}^{ – 5}}{left( {frac{{{{13}^2}}}{{{{13}^8}}}} right)^{ – 1}} = {13^{ – 5}}left( {{3^6}} right) hfill \
{text{1}}{{text{3}}^{ – 5}}{left( {frac{{{{13}^2}}}{{{{13}^8}}}} right)^{ – 1}} = {13^{ – 5 + left( 6 right)}} hfill \
{text{1}}{{text{3}}^{ – 5}}{left( {frac{{{{13}^2}}}{{{{13}^8}}}} right)^{ – 1}} = {13^1} hfill \
{text{1}}{{text{3}}^{ – 5}}{left( {frac{{{{13}^2}}}{{{{13}^8}}}} right)^{ – 1}} = 13 hfill \
end{gathered} ]
{text{f) Simplify the expression}}{text{.}} hfill \
{text{1}}{{text{3}}^{ – 5}}{left( {frac{{{{13}^2}}}{{{{13}^8}}}} right)^{ – 1}} = {13^{ – 5}}{left( {{{13}^{2 – 8}}} right)^{ – 1}} hfill \
{text{1}}{{text{3}}^{ – 5}}{left( {frac{{{{13}^2}}}{{{{13}^8}}}} right)^{ – 1}} = {13^{ – 5}}{left( {{{13}^{ – 6}}} right)^{ – 1}} hfill \
{text{Remember this rule, (}}{a^n}{)^m} = {a^{n cdot m}} hfill \
{text{1}}{{text{3}}^{ – 5}}{left( {frac{{{{13}^2}}}{{{{13}^8}}}} right)^{ – 1}} = {13^{ – 5}}left( {{{13}^{ – 6left( { – 1} right)}}} right) hfill \
{text{Remember this rule, }}{a^n}left( {{a^m}} right) = {a^{n + m}}{text{ }} hfill \
{text{1}}{{text{3}}^{ – 5}}{left( {frac{{{{13}^2}}}{{{{13}^8}}}} right)^{ – 1}} = {13^{ – 5}}left( {{3^6}} right) hfill \
{text{1}}{{text{3}}^{ – 5}}{left( {frac{{{{13}^2}}}{{{{13}^8}}}} right)^{ – 1}} = {13^{ – 5 + left( 6 right)}} hfill \
{text{1}}{{text{3}}^{ – 5}}{left( {frac{{{{13}^2}}}{{{{13}^8}}}} right)^{ – 1}} = {13^1} hfill \
{text{1}}{{text{3}}^{ – 5}}{left( {frac{{{{13}^2}}}{{{{13}^8}}}} right)^{ – 1}} = 13 hfill \
end{gathered} ]
Result
7 of 7
a) $10^3=1000$
b) $8^{-1}=dfrac{1}{8}$
c) $6^{-1}=dfrac{1}{6}$
d) $4^2=16$
e) $2^{-3}=dfrac{1}{8}$
f) $13^1=13$
Exercise 7
Step 1
1 of 7
[begin{gathered}
{text{a) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{16^{ – 1}} – {2^{ – 2}} = frac{1}{{{{16}^1}}} – frac{1}{{{2^{ – 2}}}} hfill \
{text{Simplify}}{text{.}} hfill \
= frac{1}{{16}} – frac{1}{4} hfill \
= frac{1}{{16}} – frac{4}{{16}} = frac{{1 – 4}}{{16}} hfill \
= – frac{3}{{16}} hfill \
end{gathered} ]
{text{a) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{16^{ – 1}} – {2^{ – 2}} = frac{1}{{{{16}^1}}} – frac{1}{{{2^{ – 2}}}} hfill \
{text{Simplify}}{text{.}} hfill \
= frac{1}{{16}} – frac{1}{4} hfill \
= frac{1}{{16}} – frac{4}{{16}} = frac{{1 – 4}}{{16}} hfill \
= – frac{3}{{16}} hfill \
end{gathered} ]
Step 2
2 of 7
[begin{gathered}
{text{b) Simplify the expression}}{text{.}} hfill \
{left( 3 right)^{ – 1}} + {left( 4 right)^0} – {6^{ – 1}} = – frac{1}{3} + 1 – frac{1}{6} hfill \
{text{Find their LCD}}{text{.}} hfill \
= – frac{2}{6} + frac{6}{6} – frac{1}{6}{text{.}} hfill \
= frac{3}{6}{text{ = }}frac{3}{{3 times 2}} = frac{1}{2} hfill \
end{gathered} ]
{text{b) Simplify the expression}}{text{.}} hfill \
{left( 3 right)^{ – 1}} + {left( 4 right)^0} – {6^{ – 1}} = – frac{1}{3} + 1 – frac{1}{6} hfill \
{text{Find their LCD}}{text{.}} hfill \
= – frac{2}{6} + frac{6}{6} – frac{1}{6}{text{.}} hfill \
= frac{3}{6}{text{ = }}frac{3}{{3 times 2}} = frac{1}{2} hfill \
end{gathered} ]
Step 3
3 of 7
[begin{gathered}
{text{c) Remember this rule, }}{left( {frac{a}{b}} right)^{ – n}} = frac{1}{{{{left( {frac{a}{b}} right)}^n}}} = {left( {frac{b}{a}} right)^n} hfill \
left( { – {{frac{2}{3}}^{ – 1}}} right) + left( {{{frac{2}{5}}^{ – 1}}} right) = left( { – {{frac{3}{2}}^1}} right) + left( {{{frac{5}{2}}^1}} right) hfill \
= – frac{3}{2} + frac{5}{2} hfill \
= frac{2}{2}{text{ or }}1 hfill \
end{gathered} ]
{text{c) Remember this rule, }}{left( {frac{a}{b}} right)^{ – n}} = frac{1}{{{{left( {frac{a}{b}} right)}^n}}} = {left( {frac{b}{a}} right)^n} hfill \
left( { – {{frac{2}{3}}^{ – 1}}} right) + left( {{{frac{2}{5}}^{ – 1}}} right) = left( { – {{frac{3}{2}}^1}} right) + left( {{{frac{5}{2}}^1}} right) hfill \
= – frac{3}{2} + frac{5}{2} hfill \
= frac{2}{2}{text{ or }}1 hfill \
end{gathered} ]
Step 4
4 of 7
[begin{gathered}
{text{d) Remember this rule, }}{left( {frac{a}{b}} right)^{ – n}} = frac{1}{{{{left( {frac{a}{b}} right)}^n}}} = {left( {frac{b}{a}} right)^n} hfill \
{left( {frac{1}{5}} right)^{ – 1}} + {left( { – frac{1}{2}} right)^{ – 2}} = {left( {frac{5}{1}} right)^1} + {left( { – frac{2}{1}} right)^2} hfill \
{text{Simplify}}{text{.}} hfill \
{left( {frac{1}{5}} right)^{ – 1}} + {left( { – frac{1}{2}} right)^{ – 2}} = 5 + 4 hfill \
{left( {frac{1}{5}} right)^{ – 1}} + {left( { – frac{1}{2}} right)^{ – 2}} = 9 hfill \
end{gathered} ]
{text{d) Remember this rule, }}{left( {frac{a}{b}} right)^{ – n}} = frac{1}{{{{left( {frac{a}{b}} right)}^n}}} = {left( {frac{b}{a}} right)^n} hfill \
{left( {frac{1}{5}} right)^{ – 1}} + {left( { – frac{1}{2}} right)^{ – 2}} = {left( {frac{5}{1}} right)^1} + {left( { – frac{2}{1}} right)^2} hfill \
{text{Simplify}}{text{.}} hfill \
{left( {frac{1}{5}} right)^{ – 1}} + {left( { – frac{1}{2}} right)^{ – 2}} = 5 + 4 hfill \
{left( {frac{1}{5}} right)^{ – 1}} + {left( { – frac{1}{2}} right)^{ – 2}} = 9 hfill \
end{gathered} ]
Step 5
5 of 7
[begin{gathered}
{text{e) Simplify the expression}}{text{.}} hfill \
{{text{5}}^{ – 3}} + {10^{ – 3}} – 8{left( {1000} right)^{ – 1}} = frac{1}{{{5^3}}} + frac{1}{{{{10}^3}}} – 8left( {frac{1}{{{{1000}^1}}}} right) hfill \
{text{Evaluate the power}}{text{.}} hfill \
= frac{1}{{125}} + frac{1}{{{{10}^3}}} – 8left( {frac{1}{{125 times 8}}} right) hfill \
hfill \
= frac{1}{{125}} + frac{1}{{{{10}^3}}} – left( {frac{1}{{125}}} right) hfill \
{text{Therefore,}} hfill \
= frac{1}{{{{10}^3}}} = frac{1}{{10 times 10{kern 1pt} times 10}} hfill \
hfill \
= frac{1}{{1000}} hfill \
end{gathered} ]
{text{e) Simplify the expression}}{text{.}} hfill \
{{text{5}}^{ – 3}} + {10^{ – 3}} – 8{left( {1000} right)^{ – 1}} = frac{1}{{{5^3}}} + frac{1}{{{{10}^3}}} – 8left( {frac{1}{{{{1000}^1}}}} right) hfill \
{text{Evaluate the power}}{text{.}} hfill \
= frac{1}{{125}} + frac{1}{{{{10}^3}}} – 8left( {frac{1}{{125 times 8}}} right) hfill \
hfill \
= frac{1}{{125}} + frac{1}{{{{10}^3}}} – left( {frac{1}{{125}}} right) hfill \
{text{Therefore,}} hfill \
= frac{1}{{{{10}^3}}} = frac{1}{{10 times 10{kern 1pt} times 10}} hfill \
hfill \
= frac{1}{{1000}} hfill \
end{gathered} ]
Step 6
6 of 7
[begin{gathered}
{text{f) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{{text{3}}^{ – 2}} – {6^{ – 2}} + frac{3}{2}{left( { – 9} right)^{ – 1}} = frac{1}{{{3^2}}} – frac{1}{{{6^2}}} + frac{3}{2}left( {frac{1}{{{{left( { – 9} right)}^1}}}} right) hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
= frac{1}{9} – frac{1}{{36}} + frac{3}{2}left( { – frac{1}{9}} right) hfill \
{text{Multiply}}{text{.}} hfill \
= frac{1}{9} – frac{1}{{36}} – frac{3}{{18}} hfill \
{text{Find their common denaminator}}{text{.}} hfill \
= frac{4}{{36}} – frac{1}{{36}} – frac{6}{{36}} hfill \
hfill \
= – frac{3}{{36}}{text{ = }} – frac{3}{{12 times 3}}{text{ = }} – frac{1}{{12}} hfill \
end{gathered} ]
{text{f) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{{text{3}}^{ – 2}} – {6^{ – 2}} + frac{3}{2}{left( { – 9} right)^{ – 1}} = frac{1}{{{3^2}}} – frac{1}{{{6^2}}} + frac{3}{2}left( {frac{1}{{{{left( { – 9} right)}^1}}}} right) hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
= frac{1}{9} – frac{1}{{36}} + frac{3}{2}left( { – frac{1}{9}} right) hfill \
{text{Multiply}}{text{.}} hfill \
= frac{1}{9} – frac{1}{{36}} – frac{3}{{18}} hfill \
{text{Find their common denaminator}}{text{.}} hfill \
= frac{4}{{36}} – frac{1}{{36}} – frac{6}{{36}} hfill \
hfill \
= – frac{3}{{36}}{text{ = }} – frac{3}{{12 times 3}}{text{ = }} – frac{1}{{12}} hfill \
end{gathered} ]
Result
7 of 7
a) $-dfrac{3}{16}$
b) $dfrac{1}{2}$
c) 1
d) 9
e) $dfrac{1}{1000}$
f) $-dfrac{1}{12}$
Exercise 8
Step 1
1 of 7
[begin{gathered}
{text{a) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{{text{5}}^2}{left( { – 10} right)^{ – 4}} = {5^2}left( {frac{1}{{ – {{10}^4}}}} right) hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
= 25left( {frac{1}{{10,000}}} right) hfill \
= frac{{25}}{{10,000}}{text{ = }}frac{{25}}{{400 times 25}}{text{ = }}frac{1}{{400}} hfill \
end{gathered} ]
{text{a) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{{text{5}}^2}{left( { – 10} right)^{ – 4}} = {5^2}left( {frac{1}{{ – {{10}^4}}}} right) hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
= 25left( {frac{1}{{10,000}}} right) hfill \
= frac{{25}}{{10,000}}{text{ = }}frac{{25}}{{400 times 25}}{text{ = }}frac{1}{{400}} hfill \
end{gathered} ]
Step 2
2 of 7
[begin{gathered}
{text{b) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{12^{ – 1}}{left( 2 right)^5} = frac{1}{{{{12}^1}}}{left( 2 right)^5} hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
= frac{1}{{12}}left( {32} right) hfill \
= frac{{32}}{{16}}{text{ = }}frac{{16, times 2}}{{16}}{text{ = 2}} hfill \
end{gathered} ]
{text{b) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{12^{ – 1}}{left( 2 right)^5} = frac{1}{{{{12}^1}}}{left( 2 right)^5} hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
= frac{1}{{12}}left( {32} right) hfill \
= frac{{32}}{{16}}{text{ = }}frac{{16, times 2}}{{16}}{text{ = 2}} hfill \
end{gathered} ]
Step 3
3 of 7
[begin{gathered}
{text{c) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
frac{{{{12}^{ – 1}}}}{{{{( – 4)}^{ – 1}}}} = frac{{frac{1}{{{{12}^1}}}}}{{frac{1}{{{{left( { – 4} right)}^1}}}}} hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
= frac{{frac{1}{{12}}}}{{frac{1}{{left( { – 4} right)}}}} hfill \
hfill \
= frac{1}{{12}} cdot frac{{left( { – 4} right)}}{1} hfill \
hfill \
= – frac{4}{{12}}{text{ or }}-frac{1}{3} hfill \
end{gathered} ]
{text{c) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
frac{{{{12}^{ – 1}}}}{{{{( – 4)}^{ – 1}}}} = frac{{frac{1}{{{{12}^1}}}}}{{frac{1}{{{{left( { – 4} right)}^1}}}}} hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
= frac{{frac{1}{{12}}}}{{frac{1}{{left( { – 4} right)}}}} hfill \
hfill \
= frac{1}{{12}} cdot frac{{left( { – 4} right)}}{1} hfill \
hfill \
= – frac{4}{{12}}{text{ or }}-frac{1}{3} hfill \
end{gathered} ]
Step 4
4 of 7
[begin{gathered}
{text{d) Simplify the expression}}{text{.}} hfill \
frac{{{{left( { – 9} right)}^{ – 2}}}}{{{{left( {{3^{ – 1}}} right)}^2}}} = frac{{frac{1}{{{{left( { – 9} right)}^2}}}}}{{frac{1}{{{3^{ – 2}}}}}} hfill \
{text{Evaluate the expression}}{text{.}} hfill \
frac{{{{left( { – 9} right)}^{ – 2}}}}{{{{left( {{3^{ – 1}}} right)}^2}}} = frac{{frac{1}{{left( {81} right)}}}}{{frac{1}{9}}} hfill \
hfill \
frac{{{{left( { – 9} right)}^{ – 2}}}}{{{{left( {{3^{ – 1}}} right)}^2}}} = frac{1}{{left( {81} right)}} cdot frac{9}{1} hfill \
hfill \
frac{{{{left( { – 9} right)}^{ – 2}}}}{{{{left( {{3^{ – 1}}} right)}^2}}} = frac{9}{{81}}{text{ }} = frac{9}{{9 times 9}} = frac{1}{9} hfill \
end{gathered} ]
{text{d) Simplify the expression}}{text{.}} hfill \
frac{{{{left( { – 9} right)}^{ – 2}}}}{{{{left( {{3^{ – 1}}} right)}^2}}} = frac{{frac{1}{{{{left( { – 9} right)}^2}}}}}{{frac{1}{{{3^{ – 2}}}}}} hfill \
{text{Evaluate the expression}}{text{.}} hfill \
frac{{{{left( { – 9} right)}^{ – 2}}}}{{{{left( {{3^{ – 1}}} right)}^2}}} = frac{{frac{1}{{left( {81} right)}}}}{{frac{1}{9}}} hfill \
hfill \
frac{{{{left( { – 9} right)}^{ – 2}}}}{{{{left( {{3^{ – 1}}} right)}^2}}} = frac{1}{{left( {81} right)}} cdot frac{9}{1} hfill \
hfill \
frac{{{{left( { – 9} right)}^{ – 2}}}}{{{{left( {{3^{ – 1}}} right)}^2}}} = frac{9}{{81}}{text{ }} = frac{9}{{9 times 9}} = frac{1}{9} hfill \
end{gathered} ]
Step 5
5 of 7
[begin{gathered}
{text{e)}} hfill \
{{text{8}}^{ – 1}}left( {frac{{{2^{ – 3}}}}{{{4^{ – 1}}}}} right) = {left( {frac{8}{4}} right)^{ – 1}} cdot {2^{ – 3}} hfill \
= {2^{ – 1}} cdot {2^{ – 3}} hfill \
= {2^{ – 1 + left( { – 3} right)}} hfill \
= {2^{ – 4}} hfill \
{text{Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{{text{8}}^{ – 1}}left( {frac{{{2^{ – 3}}}}{{{4^{ – 1}}}}} right) = frac{1}{{{2^4}}}{text{ or }}frac{1}{{16}} hfill \
end{gathered} ]
{text{e)}} hfill \
{{text{8}}^{ – 1}}left( {frac{{{2^{ – 3}}}}{{{4^{ – 1}}}}} right) = {left( {frac{8}{4}} right)^{ – 1}} cdot {2^{ – 3}} hfill \
= {2^{ – 1}} cdot {2^{ – 3}} hfill \
= {2^{ – 1 + left( { – 3} right)}} hfill \
= {2^{ – 4}} hfill \
{text{Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{{text{8}}^{ – 1}}left( {frac{{{2^{ – 3}}}}{{{4^{ – 1}}}}} right) = frac{1}{{{2^4}}}{text{ or }}frac{1}{{16}} hfill \
end{gathered} ]
Step 6
6 of 7
[begin{gathered}
{text{f) }} hfill \
frac{{{{left( { – 5} right)}^3}{{left( { – 25} right)}^{ – 1}}}}{{{{left( { – 5} right)}^{ – 2}}}} = frac{{left( {{5^3} cdot {5^2}} right)}}{{left( {25} right)}} hfill \
frac{{{{left( { – 5} right)}^3}{{left( { – 25} right)}^{ – 1}}}}{{{{left( { – 5} right)}^{ – 2}}}} = frac{{{5^3} cdot {5^2}}}{{{5^2}}} hfill \
{text{Remember this rule, }}{{text{a}}^n} div {a^m} = {a^{n – m}} hfill \
frac{{{{left( { – 5} right)}^3}{{left( { – 25} right)}^{ – 1}}}}{{{{left( { – 5} right)}^{ – 2}}}} = frac{{{5^5}}}{{{5^2}}} hfill \
frac{{{{left( { – 5} right)}^3}{{left( { – 25} right)}^{ – 1}}}}{{{{left( { – 5} right)}^{ – 2}}}} = {5^3} hfill \
{text{Evaluate the exponents}}{text{.}} hfill \
frac{{{{left( { – 5} right)}^3}{{left( { – 25} right)}^{ – 1}}}}{{{{left( { – 5} right)}^{ – 2}}}} = 125 hfill \
end{gathered} ]
{text{f) }} hfill \
frac{{{{left( { – 5} right)}^3}{{left( { – 25} right)}^{ – 1}}}}{{{{left( { – 5} right)}^{ – 2}}}} = frac{{left( {{5^3} cdot {5^2}} right)}}{{left( {25} right)}} hfill \
frac{{{{left( { – 5} right)}^3}{{left( { – 25} right)}^{ – 1}}}}{{{{left( { – 5} right)}^{ – 2}}}} = frac{{{5^3} cdot {5^2}}}{{{5^2}}} hfill \
{text{Remember this rule, }}{{text{a}}^n} div {a^m} = {a^{n – m}} hfill \
frac{{{{left( { – 5} right)}^3}{{left( { – 25} right)}^{ – 1}}}}{{{{left( { – 5} right)}^{ – 2}}}} = frac{{{5^5}}}{{{5^2}}} hfill \
frac{{{{left( { – 5} right)}^3}{{left( { – 25} right)}^{ – 1}}}}{{{{left( { – 5} right)}^{ – 2}}}} = {5^3} hfill \
{text{Evaluate the exponents}}{text{.}} hfill \
frac{{{{left( { – 5} right)}^3}{{left( { – 25} right)}^{ – 1}}}}{{{{left( { – 5} right)}^{ – 2}}}} = 125 hfill \
end{gathered} ]
Result
7 of 7
a) $dfrac{1}{400}$
b) 2
c) $-dfrac{1}{3}$
d) $dfrac{1}{9}$
e) $dfrac{1}{16}$
f) 125
Exercise 9
Step 1
1 of 2
[begin{gathered}
{text{a) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{left( { – 4} right)^{ – 3}} = frac{1}{{{{left( { – 4} right)}^3}}} hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
{left( { – 4} right)^{ – 3}} = – frac{1}{{64}} hfill \
hfill \
{text{b) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{left( { – 4} right)^{ – 2}} = frac{1}{{{{left( { – 4} right)}^2}}} hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
{left( { – 4} right)^{ – 2}} = frac{1}{{ – 16}}{text{ or}} – frac{1}{{16}} hfill \
hfill \
{text{c)Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
– {left( 5 right)^{ – 3}} = – frac{1}{{{5^3}}} hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
– {left( 5 right)^{ – 3}} = frac{1}{{125}} hfill \
{text{d) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
– {left( 5 right)^{ – 2}} = – frac{1}{{{5^2}}} hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
– {left( 5 right)^{ – 2}} = frac{1}{{25}} hfill \
hfill \
{text{e) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{left( { – 6} right)^{ – 3}} = frac{1}{{{{left( { – 6} right)}^3}}} hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
{left( { – 6} right)^{ – 3}} = frac{1}{{ – 126}}{text{ or }} – frac{1}{{126}} hfill \
hfill \
{text{f) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
– {left( 6 right)^{ – 2}} = – frac{1}{{{6^2}}} hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
– {left( 6 right)^{ – 2}} = – frac{1}{{36}} hfill \
hfill \
end{gathered} ]
{text{a) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{left( { – 4} right)^{ – 3}} = frac{1}{{{{left( { – 4} right)}^3}}} hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
{left( { – 4} right)^{ – 3}} = – frac{1}{{64}} hfill \
hfill \
{text{b) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{left( { – 4} right)^{ – 2}} = frac{1}{{{{left( { – 4} right)}^2}}} hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
{left( { – 4} right)^{ – 2}} = frac{1}{{ – 16}}{text{ or}} – frac{1}{{16}} hfill \
hfill \
{text{c)Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
– {left( 5 right)^{ – 3}} = – frac{1}{{{5^3}}} hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
– {left( 5 right)^{ – 3}} = frac{1}{{125}} hfill \
{text{d) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
– {left( 5 right)^{ – 2}} = – frac{1}{{{5^2}}} hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
– {left( 5 right)^{ – 2}} = frac{1}{{25}} hfill \
hfill \
{text{e) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{left( { – 6} right)^{ – 3}} = frac{1}{{{{left( { – 6} right)}^3}}} hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
{left( { – 6} right)^{ – 3}} = frac{1}{{ – 126}}{text{ or }} – frac{1}{{126}} hfill \
hfill \
{text{f) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
– {left( 6 right)^{ – 2}} = – frac{1}{{{6^2}}} hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
– {left( 6 right)^{ – 2}} = – frac{1}{{36}} hfill \
hfill \
end{gathered} ]
Result
2 of 2
a) $-dfrac{1}{64}$
b) $dfrac{1}{16}$
c) $-dfrac{1}{125}$
d) $-dfrac{1}{25}$
e) $-dfrac{1}{216}$
f) $-dfrac{1}{36}$
Exercise 10
Step 1
1 of 2
[begin{gathered}
{text{The larger the denominator the smaller the number}}{text{.}} hfill \
{text{Since }} hfill \
{{text{5}}^{ – 2}} = frac{1}{{25}} hfill \
{10^{ – 1}} = frac{1}{{10}} hfill \
{text{ }}{3^{ – 2}} = frac{1}{9} hfill \
{2^{ – 3}} = frac{1}{8} hfill \
{text{ }}{4^{ – 1}} = frac{1}{4} hfill \
{left( {0.1} right)^{ – 1}} = frac{1}{{0.01}} hfill \
{text{The numbers order from least to greatest are }}{{text{5}}^{ – 2}},{10^{ – 1}},{3^{ – 2}},{2^{ – 3}},{4^{ – 1}},{left( {0.1} right)^{ – 1}}{text{. }} hfill \
end{gathered} ]
{text{The larger the denominator the smaller the number}}{text{.}} hfill \
{text{Since }} hfill \
{{text{5}}^{ – 2}} = frac{1}{{25}} hfill \
{10^{ – 1}} = frac{1}{{10}} hfill \
{text{ }}{3^{ – 2}} = frac{1}{9} hfill \
{2^{ – 3}} = frac{1}{8} hfill \
{text{ }}{4^{ – 1}} = frac{1}{4} hfill \
{left( {0.1} right)^{ – 1}} = frac{1}{{0.01}} hfill \
{text{The numbers order from least to greatest are }}{{text{5}}^{ – 2}},{10^{ – 1}},{3^{ – 2}},{2^{ – 3}},{4^{ – 1}},{left( {0.1} right)^{ – 1}}{text{. }} hfill \
end{gathered} ]
Result
2 of 2
$$
{{text{5}}^{ – 2}},{10^{ – 1}},{3^{ – 2}},{2^{ – 3}},{4^{ – 1}},{left( {0.1} right)^{ – 1}}
$$
{{text{5}}^{ – 2}},{10^{ – 1}},{3^{ – 2}},{2^{ – 3}},{4^{ – 1}},{left( {0.1} right)^{ – 1}}
$$
Exercise 11
Step 1
1 of 2
[begin{gathered}
{text{In this excercise we shall substitute }}x = – 2,{text{ }}y = 3{text{ , }}n = – 1 hfill \
hfill \
{text{a) Substitute the numbers}}{text{.}} hfill \
{left( {{x^n} + {y^n}} right)^{ – 2n}} = {left[ {left( { – {2^{ – 1}}} right) + {3^{ – 1}}} right]^{ – 2left( { – 1} right)}} hfill \
= {left[ {left( { – {2^{ – 1}}} right) + {3^{ – 1}}} right]^2} hfill \
{text{Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
= {left( { – {2^{ – 1}} + {3^{ – 1}}} right)^2} hfill \
= {left( {frac{1}{{ – 2}} + frac{1}{3}} right)^2} hfill \
{text{Add}}{text{.}} hfill \
= {left( { – frac{3}{6} + frac{2}{6}} right)^2} hfill \
= {left( {frac{1}{6}} right)^2} hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
= left( {frac{1}{{36}}} right) hfill \
hfill \
{text{b) Substitute the numbers}}{text{.}} hfill \
{left( {{x^2}} right)^n}left( {{y^{ – 2n}}} right){x^{ – n}} = {left( { – {2^2}} right)^{ – 1}}left( {{3^{ – 2left( { – 1} right)}}} right) – {left( 2 right)^1} hfill \
{text{Multiply and evaluate the power}}{text{.}} hfill \
= left( { – {2^{ – 2}}} right)left( {{3^2}} right){left( { – 2} right)^1} hfill \
= left( { – {2^{ – 2 + 1}}} right)left( 9 right) hfill \
= left( { – {2^{ – 1}}} right)left( 9 right) hfill \
= left( {frac{1}{{ – {2^1}}}} right)left( 9 right) hfill \
= – frac{9}{2} hfill \
hfill \
{text{c) Substitute the numbers}}{text{.}} hfill \
{left( {frac{{{x^n}}}{{{y^n}}}} right)^n} = {left( {frac{{left( { – {2^{ – 1}}} right)}}{{left( {{3^{ – 1}}} right)}}} right)^{ – 1}} hfill \
{text{Remember this rule, }}{left( {frac{a}{b}} right)^{ – n}} = frac{1}{{{{left( {frac{a}{b}} right)}^n}}} = {left( {frac{b}{a}} right)^n} hfill \
{left( {frac{{{x^n}}}{{{y^n}}}} right)^n} = frac{{{3^{ – 1}}}}{{{{left( { – 2} right)}^{ – 1}}}} hfill \
{text{Move the expression to the denaminator and }} hfill \
{text{make the exponent positive}} hfill \
= frac{{ – 2}}{3}{text{ or }} – frac{2}{3} hfill \
hfill \
{text{d) Substitute the numbers}}{text{.}} hfill \
{left( {frac{{left( {x{y^n}} right)}}{{{{left( {xy} right)}^{2n}}}}} right)^{2n}} = {left( {frac{{left( {left( { – 2} right){{left( 3 right)}^{ – 1}}} right)}}{{{{left( { – 2left( 3 right)} right)}^{2left( { – 1} right)}}}}} right)^{2left( { – 1} right)}} hfill \
= {left( {frac{{left( {left( { – 2} right){{left( 3 right)}^{ – 1}}} right)}}{{{{left( { – 6} right)}^{ – 2}}}}} right)^{ – 2}} hfill \
{text{Move the expression to the denaminator and }} hfill \
{text{make the exponent positive}}{text{.}} hfill \
{left( {frac{{left( {x{y^n}} right)}}{{{{left( {xy} right)}^{2n}}}}} right)^{2n}} = {left( {frac{{{{left( { – 6} right)}^{ – 2}}}}{{left( {left( { – 2} right){{left( 3 right)}^{ – 1}}} right)}}} right)^2} hfill \
= {left( {frac{{{{left( 3 right)}^1}}}{{left( {left( { – 2} right){{left( { – 6} right)}^2}} right)}}} right)^2} hfill \
{text{Evaluate the exponents}}{text{.}} hfill \
= {left( {frac{{left( 3 right)}}{{left( {72} right)}}} right)^2} hfill \
= left( {frac{1}{{576}}} right) hfill \
hfill \
end{gathered} ]
{text{In this excercise we shall substitute }}x = – 2,{text{ }}y = 3{text{ , }}n = – 1 hfill \
hfill \
{text{a) Substitute the numbers}}{text{.}} hfill \
{left( {{x^n} + {y^n}} right)^{ – 2n}} = {left[ {left( { – {2^{ – 1}}} right) + {3^{ – 1}}} right]^{ – 2left( { – 1} right)}} hfill \
= {left[ {left( { – {2^{ – 1}}} right) + {3^{ – 1}}} right]^2} hfill \
{text{Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
= {left( { – {2^{ – 1}} + {3^{ – 1}}} right)^2} hfill \
= {left( {frac{1}{{ – 2}} + frac{1}{3}} right)^2} hfill \
{text{Add}}{text{.}} hfill \
= {left( { – frac{3}{6} + frac{2}{6}} right)^2} hfill \
= {left( {frac{1}{6}} right)^2} hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
= left( {frac{1}{{36}}} right) hfill \
hfill \
{text{b) Substitute the numbers}}{text{.}} hfill \
{left( {{x^2}} right)^n}left( {{y^{ – 2n}}} right){x^{ – n}} = {left( { – {2^2}} right)^{ – 1}}left( {{3^{ – 2left( { – 1} right)}}} right) – {left( 2 right)^1} hfill \
{text{Multiply and evaluate the power}}{text{.}} hfill \
= left( { – {2^{ – 2}}} right)left( {{3^2}} right){left( { – 2} right)^1} hfill \
= left( { – {2^{ – 2 + 1}}} right)left( 9 right) hfill \
= left( { – {2^{ – 1}}} right)left( 9 right) hfill \
= left( {frac{1}{{ – {2^1}}}} right)left( 9 right) hfill \
= – frac{9}{2} hfill \
hfill \
{text{c) Substitute the numbers}}{text{.}} hfill \
{left( {frac{{{x^n}}}{{{y^n}}}} right)^n} = {left( {frac{{left( { – {2^{ – 1}}} right)}}{{left( {{3^{ – 1}}} right)}}} right)^{ – 1}} hfill \
{text{Remember this rule, }}{left( {frac{a}{b}} right)^{ – n}} = frac{1}{{{{left( {frac{a}{b}} right)}^n}}} = {left( {frac{b}{a}} right)^n} hfill \
{left( {frac{{{x^n}}}{{{y^n}}}} right)^n} = frac{{{3^{ – 1}}}}{{{{left( { – 2} right)}^{ – 1}}}} hfill \
{text{Move the expression to the denaminator and }} hfill \
{text{make the exponent positive}} hfill \
= frac{{ – 2}}{3}{text{ or }} – frac{2}{3} hfill \
hfill \
{text{d) Substitute the numbers}}{text{.}} hfill \
{left( {frac{{left( {x{y^n}} right)}}{{{{left( {xy} right)}^{2n}}}}} right)^{2n}} = {left( {frac{{left( {left( { – 2} right){{left( 3 right)}^{ – 1}}} right)}}{{{{left( { – 2left( 3 right)} right)}^{2left( { – 1} right)}}}}} right)^{2left( { – 1} right)}} hfill \
= {left( {frac{{left( {left( { – 2} right){{left( 3 right)}^{ – 1}}} right)}}{{{{left( { – 6} right)}^{ – 2}}}}} right)^{ – 2}} hfill \
{text{Move the expression to the denaminator and }} hfill \
{text{make the exponent positive}}{text{.}} hfill \
{left( {frac{{left( {x{y^n}} right)}}{{{{left( {xy} right)}^{2n}}}}} right)^{2n}} = {left( {frac{{{{left( { – 6} right)}^{ – 2}}}}{{left( {left( { – 2} right){{left( 3 right)}^{ – 1}}} right)}}} right)^2} hfill \
= {left( {frac{{{{left( 3 right)}^1}}}{{left( {left( { – 2} right){{left( { – 6} right)}^2}} right)}}} right)^2} hfill \
{text{Evaluate the exponents}}{text{.}} hfill \
= {left( {frac{{left( 3 right)}}{{left( {72} right)}}} right)^2} hfill \
= left( {frac{1}{{576}}} right) hfill \
hfill \
end{gathered} ]
Result
2 of 2
a) $dfrac{1}{36}$
b) $-dfrac{9}{2}$
c) $-dfrac{2}{3}$
d) $dfrac{1}{576}$
Exercise 12
Step 1
1 of 2
[begin{gathered}
{text{a) Erick did not follow this rule , }}{a^{ – n}} = frac{1}{{{a^n}}}{text{ while }} hfill \
{text{Vinn did not add the exponents correctly}}{text{.}} hfill \
hfill \
{text{b) Remember this rule, }}{a^n}left( {{a^m}} right) = {a^{n + m}}{text{ }} hfill \
{{text{3}}^{ – 2}}left( 3 right) = {3^{ – 2 + 1}} hfill \
{{text{3}}^{ – 2}}left( 3 right) = {3^{ – 1}} hfill \
{text{Remember this rule , }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{{text{3}}^{ – 2}}left( 3 right) = frac{1}{{{3^1}}}{text{ or }}frac{1}{3} hfill \
end{gathered} ]
{text{a) Erick did not follow this rule , }}{a^{ – n}} = frac{1}{{{a^n}}}{text{ while }} hfill \
{text{Vinn did not add the exponents correctly}}{text{.}} hfill \
hfill \
{text{b) Remember this rule, }}{a^n}left( {{a^m}} right) = {a^{n + m}}{text{ }} hfill \
{{text{3}}^{ – 2}}left( 3 right) = {3^{ – 2 + 1}} hfill \
{{text{3}}^{ – 2}}left( 3 right) = {3^{ – 1}} hfill \
{text{Remember this rule , }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{{text{3}}^{ – 2}}left( 3 right) = frac{1}{{{3^1}}}{text{ or }}frac{1}{3} hfill \
end{gathered} ]
Result
2 of 2
a) Erik: $3^{-1}neq -dfrac{1}{3}$
Vinn: $3^{-2}times 3^{1}neq 3^{-2}$
b) Correct Solution:
$3^{-2}times 3^{1}$
$=3^{-2+1}$
$=3^{-1}$
$$
=dfrac{1}{3}
$$
Exercise 13
Step 1
1 of 3
[begin{gathered}
{text{a) Evaluate the exponents}}{text{.}} hfill \
{2^3}left( {{4^{ – 2}}} right) div {2^2} = 8left( {frac{1}{{{4^2}}}} right) div 4 hfill \
{2^3}left( {{4^{ – 2}}} right) div {2^2} = frac{8}{{16}} div 4 hfill \
{2^3}left( {{4^{ – 2}}} right) div {2^2} = frac{1}{8} hfill \
hfill \
{text{b) Multiply}}{text{.}} hfill \
{left( {2 cdot 3} right)^{ – 1}} = {left( 6 right)^{ – 1}} hfill \
{text{Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{left( {2 cdot 3} right)^{ – 1}} = frac{1}{{{{left( 6 right)}^1}}}{text{ or }}frac{1}{6} hfill \
hfill \
{text{c) Remember this rule, }}{left( {frac{a}{b}} right)^{ – n}} = frac{1}{{{{left( {frac{a}{b}} right)}^n}}} = {left( {frac{b}{a}} right)^n} hfill \
{left( {frac{{{3^1}}}{{{2^1}}}} right)^{ – 2}} = {left( {frac{{{2^{ – 1}}}}{{{3^{ – 1}}}}} right)^2} hfill \
{text{Express with a positive exponent}}{text{.}} hfill \
{left( {frac{{{3^1}}}{{{2^1}}}} right)^{ – 2}} = {left( {frac{3}{2}} right)^2} hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
{left( {frac{{{3^1}}}{{{2^1}}}} right)^{ – 2}} = frac{9}{4} hfill \
hfill \
{text{d) Evaluate the exponent}}{text{.}} hfill \
{4^{ – 1}}left( {{4^2} + {4^0}} right) = frac{1}{{{4^1}}}left( {16 + 1} right) hfill \
{text{Add}}{text{.}} hfill \
{4^{ – 1}}left( {{4^2} + {4^0}} right) = frac{1}{4}left( {17} right) hfill \
{text{Multiply}}{text{.}} hfill \
{4^{ – 1}}left( {{4^2} + {4^0}} right) = frac{{17}}{4} hfill \
hfill \
{text{e) Reduce the expression with }}{{text{2}}^4}{text{ and }}{3^{ – 2}} hfill \
frac{{{2^5}}}{{{3^{ – 2}}}} times frac{{{3^1}}}{{{2^4}}} = 2 times 3 hfill \
{text{Multiply}}{text{.}} hfill \
frac{{{2^5}}}{{{3^{ – 2}}}} times frac{{{3^1}}}{{{2^4}}} = 6 hfill \
hfill \
{text{f) Evaluate the exponents}}{text{.}} hfill \
{left( {{5^0} + {5^2}} right)^{ – 1}} = {left( {1 + 25} right)^{ – 1}} hfill \
{left( {{5^0} + {5^2}} right)^{ – 1}} = {left( {26} right)^{ – 1}} hfill \
{text{Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{left( {{5^0} + {5^2}} right)^{ – 1}} = frac{1}{{26}} hfill \
end{gathered} ]
{text{a) Evaluate the exponents}}{text{.}} hfill \
{2^3}left( {{4^{ – 2}}} right) div {2^2} = 8left( {frac{1}{{{4^2}}}} right) div 4 hfill \
{2^3}left( {{4^{ – 2}}} right) div {2^2} = frac{8}{{16}} div 4 hfill \
{2^3}left( {{4^{ – 2}}} right) div {2^2} = frac{1}{8} hfill \
hfill \
{text{b) Multiply}}{text{.}} hfill \
{left( {2 cdot 3} right)^{ – 1}} = {left( 6 right)^{ – 1}} hfill \
{text{Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{left( {2 cdot 3} right)^{ – 1}} = frac{1}{{{{left( 6 right)}^1}}}{text{ or }}frac{1}{6} hfill \
hfill \
{text{c) Remember this rule, }}{left( {frac{a}{b}} right)^{ – n}} = frac{1}{{{{left( {frac{a}{b}} right)}^n}}} = {left( {frac{b}{a}} right)^n} hfill \
{left( {frac{{{3^1}}}{{{2^1}}}} right)^{ – 2}} = {left( {frac{{{2^{ – 1}}}}{{{3^{ – 1}}}}} right)^2} hfill \
{text{Express with a positive exponent}}{text{.}} hfill \
{left( {frac{{{3^1}}}{{{2^1}}}} right)^{ – 2}} = {left( {frac{3}{2}} right)^2} hfill \
{text{Evaluate the exponent}}{text{.}} hfill \
{left( {frac{{{3^1}}}{{{2^1}}}} right)^{ – 2}} = frac{9}{4} hfill \
hfill \
{text{d) Evaluate the exponent}}{text{.}} hfill \
{4^{ – 1}}left( {{4^2} + {4^0}} right) = frac{1}{{{4^1}}}left( {16 + 1} right) hfill \
{text{Add}}{text{.}} hfill \
{4^{ – 1}}left( {{4^2} + {4^0}} right) = frac{1}{4}left( {17} right) hfill \
{text{Multiply}}{text{.}} hfill \
{4^{ – 1}}left( {{4^2} + {4^0}} right) = frac{{17}}{4} hfill \
hfill \
{text{e) Reduce the expression with }}{{text{2}}^4}{text{ and }}{3^{ – 2}} hfill \
frac{{{2^5}}}{{{3^{ – 2}}}} times frac{{{3^1}}}{{{2^4}}} = 2 times 3 hfill \
{text{Multiply}}{text{.}} hfill \
frac{{{2^5}}}{{{3^{ – 2}}}} times frac{{{3^1}}}{{{2^4}}} = 6 hfill \
hfill \
{text{f) Evaluate the exponents}}{text{.}} hfill \
{left( {{5^0} + {5^2}} right)^{ – 1}} = {left( {1 + 25} right)^{ – 1}} hfill \
{left( {{5^0} + {5^2}} right)^{ – 1}} = {left( {26} right)^{ – 1}} hfill \
{text{Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
{left( {{5^0} + {5^2}} right)^{ – 1}} = frac{1}{{26}} hfill \
end{gathered} ]
Step 2
2 of 3
[begin{gathered}
{text{g) Divide first the expression}}{text{.}} hfill \
{text{Remember this rule, }}{{text{a}}^n} div {a^m} = {a^{n – m}} hfill \
frac{{{3^{ – 2}} times {2^{ – 3}}}}{{{3^{ – 1}} times {2^{ – 2}}}} = {3^{ – 2 – left( { – 1} right)}} times {2^{ – 3 – left( { – 2} right)}} hfill \
frac{{{3^{ – 2}} times {2^{ – 3}}}}{{{3^{ – 1}} times {2^{ – 2}}}} = {3^{ – 1}} times {2^{ – 1}} hfill \
{text{Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
frac{{{3^{ – 2}} times {2^{ – 3}}}}{{{3^{ – 1}} times {2^{ – 2}}}} = frac{1}{3} times frac{1}{2} hfill \
frac{{{3^{ – 2}} times {2^{ – 3}}}}{{{3^{ – 1}} times {2^{ – 2}}}} = frac{1}{6} hfill \
hfill \
{text{h) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
frac{{{4^{ – 2}} + {3^{ – 1}}}}{{{3^{ – 2}} + {2^{ – 3}}}} = frac{{frac{1}{{{4^2}}} + frac{1}{{{3^1}}}}}{{frac{1}{{{3^2}}} + frac{1}{{{2^3}}}}} hfill \
{text{Add and then divide}}{text{.}} hfill \
frac{{{4^{ – 2}} + {3^{ – 1}}}}{{{3^{ – 2}} + {2^{ – 3}}}} = frac{{frac{1}{{16}} + frac{1}{3}}}{{frac{1}{9} + frac{1}{8}}} hfill \
frac{{frac{1}{{{4^2}}} + frac{1}{{{3^1}}}}}{{frac{1}{{{3^2}}} + frac{1}{{{2^3}}}}} = frac{{19}}{{48}} div frac{{17}}{{72}} hfill \
{text{Reciprocate the second fraction and then multiply}}{text{.}} hfill \
frac{{frac{1}{{{4^2}}} + frac{1}{{{3^1}}}}}{{frac{1}{{{3^2}}} + frac{1}{{{2^3}}}}} = frac{{19}}{{48}} times frac{{72}}{{17}} hfill \
frac{{frac{1}{{{4^2}}} + frac{1}{{{3^1}}}}}{{frac{1}{{{3^2}}} + frac{1}{{{2^3}}}}} = frac{{57}}{{34}} hfill \
hfill \
{text{i) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
frac{{{{text{5}}^{ – 1}} – {2^{ – 2}}}}{{{{text{5}}^{ – 1}} + {2^{ – 2}}}} = frac{{frac{1}{5} – frac{1}{{{2^2}}}}}{{frac{1}{5} + frac{1}{{{2^2}}}}} hfill \
{text{Evaluate the powers}} hfill \
frac{{{{text{5}}^{ – 1}} – {2^{ – 2}}}}{{{{text{5}}^{ – 1}} + {2^{ – 2}}}} = frac{{frac{1}{5} – frac{1}{4}}}{{frac{1}{5} + frac{1}{4}}} hfill \
{text{Simplify the complex fraction}} hfill \
frac{{frac{1}{5} – frac{1}{{{2^2}}}}}{{frac{1}{5} + frac{1}{{{2^2}}}}} = – frac{1}{{20}} div frac{9}{{20}} hfill \
{text{Reciprocate the second fraction and then multiply}} hfill \
frac{{frac{1}{5} – frac{1}{{{2^2}}}}}{{frac{1}{5} + frac{1}{{{2^2}}}}} = – frac{1}{{20}} times frac{{20}}{9} hfill \
frac{{frac{1}{5} – frac{1}{{{2^2}}}}}{{frac{1}{5} + frac{1}{{{2^2}}}}} = – frac{1}{9} hfill \
hfill \
end{gathered} ]
{text{g) Divide first the expression}}{text{.}} hfill \
{text{Remember this rule, }}{{text{a}}^n} div {a^m} = {a^{n – m}} hfill \
frac{{{3^{ – 2}} times {2^{ – 3}}}}{{{3^{ – 1}} times {2^{ – 2}}}} = {3^{ – 2 – left( { – 1} right)}} times {2^{ – 3 – left( { – 2} right)}} hfill \
frac{{{3^{ – 2}} times {2^{ – 3}}}}{{{3^{ – 1}} times {2^{ – 2}}}} = {3^{ – 1}} times {2^{ – 1}} hfill \
{text{Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
frac{{{3^{ – 2}} times {2^{ – 3}}}}{{{3^{ – 1}} times {2^{ – 2}}}} = frac{1}{3} times frac{1}{2} hfill \
frac{{{3^{ – 2}} times {2^{ – 3}}}}{{{3^{ – 1}} times {2^{ – 2}}}} = frac{1}{6} hfill \
hfill \
{text{h) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
frac{{{4^{ – 2}} + {3^{ – 1}}}}{{{3^{ – 2}} + {2^{ – 3}}}} = frac{{frac{1}{{{4^2}}} + frac{1}{{{3^1}}}}}{{frac{1}{{{3^2}}} + frac{1}{{{2^3}}}}} hfill \
{text{Add and then divide}}{text{.}} hfill \
frac{{{4^{ – 2}} + {3^{ – 1}}}}{{{3^{ – 2}} + {2^{ – 3}}}} = frac{{frac{1}{{16}} + frac{1}{3}}}{{frac{1}{9} + frac{1}{8}}} hfill \
frac{{frac{1}{{{4^2}}} + frac{1}{{{3^1}}}}}{{frac{1}{{{3^2}}} + frac{1}{{{2^3}}}}} = frac{{19}}{{48}} div frac{{17}}{{72}} hfill \
{text{Reciprocate the second fraction and then multiply}}{text{.}} hfill \
frac{{frac{1}{{{4^2}}} + frac{1}{{{3^1}}}}}{{frac{1}{{{3^2}}} + frac{1}{{{2^3}}}}} = frac{{19}}{{48}} times frac{{72}}{{17}} hfill \
frac{{frac{1}{{{4^2}}} + frac{1}{{{3^1}}}}}{{frac{1}{{{3^2}}} + frac{1}{{{2^3}}}}} = frac{{57}}{{34}} hfill \
hfill \
{text{i) Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
frac{{{{text{5}}^{ – 1}} – {2^{ – 2}}}}{{{{text{5}}^{ – 1}} + {2^{ – 2}}}} = frac{{frac{1}{5} – frac{1}{{{2^2}}}}}{{frac{1}{5} + frac{1}{{{2^2}}}}} hfill \
{text{Evaluate the powers}} hfill \
frac{{{{text{5}}^{ – 1}} – {2^{ – 2}}}}{{{{text{5}}^{ – 1}} + {2^{ – 2}}}} = frac{{frac{1}{5} – frac{1}{4}}}{{frac{1}{5} + frac{1}{4}}} hfill \
{text{Simplify the complex fraction}} hfill \
frac{{frac{1}{5} – frac{1}{{{2^2}}}}}{{frac{1}{5} + frac{1}{{{2^2}}}}} = – frac{1}{{20}} div frac{9}{{20}} hfill \
{text{Reciprocate the second fraction and then multiply}} hfill \
frac{{frac{1}{5} – frac{1}{{{2^2}}}}}{{frac{1}{5} + frac{1}{{{2^2}}}}} = – frac{1}{{20}} times frac{{20}}{9} hfill \
frac{{frac{1}{5} – frac{1}{{{2^2}}}}}{{frac{1}{5} + frac{1}{{{2^2}}}}} = – frac{1}{9} hfill \
hfill \
end{gathered} ]
Result
3 of 3
a) $dfrac{1}{8}$
b) $dfrac{1}{6}$
c) $dfrac{9}{4}$
d) $dfrac{17}{4}$
e) $6$
f) $dfrac{1}{26}$
g) $dfrac{1}{6}$
h) $dfrac{57}{34}$
i) $-dfrac{1}{9}$
Exercise 14
Step 1
1 of 2
[begin{gathered}
{text{In this excercise we shall substitute the following numbers}} hfill \
a = 1,{text{ }}b = 3,{text{ }}c = 2 hfill \
{text{a) }}a{c^c} = 1{left( 2 right)^2} hfill \
{text{Evaluate the power and then multiply}} hfill \
= 1left( 4 right) hfill \
= 4 hfill \
hfill \
{text{b) }}{a^c}{b^c} = {1^2}left( {{3^2}} right) hfill \
{text{Evaluate the power and then multiply}} hfill \
= 1left( 9 right) hfill \
= 9 hfill \
hfill \
{text{c) }}{left( {ab} right)^{ – c}} = {left[ {left( 1 right)left( 3 right)} right]^{ – 2}} hfill \
{text{Multiply and evaluate the power}} hfill \
= {left( 3 right)^{ – 2}} hfill \
{text{Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
= frac{1}{{{3^2}}} hfill \
= frac{1}{9} hfill \
hfill \
{text{d) }}{left( {b div c} right)^{ – a}} = {left( {3 div 2} right)^{ – 1}} hfill \
= {left( {frac{3}{2}} right)^{ – 1}} hfill \
{text{Remember this rule, }}{left( {frac{a}{b}} right)^{ – n}} = frac{1}{{{{left( {frac{a}{b}} right)}^n}}} = {left( {frac{b}{a}} right)^n} hfill \
= left( {frac{2}{3}} right) hfill \
hfill \
{text{e) }}{left( { – a div b} right)^{ – c}} = {left( { – 1 div 3} right)^{ – 2}} hfill \
= {left( {frac{{ – 1}}{3}} right)^{ – 2}} hfill \
{text{Remember this rule, }}{left( {frac{a}{b}} right)^{ – n}} = frac{1}{{{{left( {frac{a}{b}} right)}^n}}} = {left( {frac{b}{a}} right)^n} hfill \
= {left( {frac{3}{{ – 1}}} right)^2} = {left( { – 3} right)^2} hfill \
{text{Negative numbers raised to an even power is always positive}} hfill \
= 9 hfill \
hfill \
{text{f) }}{left( {{a^{ – 1}}{b^{ – 2}}} right)^c} = {left[ {left( {{1^{ – 1}}} right)left( {{3^{ – 2}}} right)} right]^2} hfill \
{text{Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
= {left( {frac{1}{1} times frac{1}{{{3^2}}}} right)^2} hfill \
{text{Evaluate and simplify the expression}} hfill \
= {left( {1 times frac{1}{9}} right)^2} hfill \
= {left( {frac{1}{9}} right)^2} hfill \
= frac{1}{{81}} hfill \
hfill \
{text{g) }}{left( {{a^b}{b^a}} right)^c} = {left( {{1^3} times {3^1}} right)^2} hfill \
{text{Multiply and evaluate the powers}} hfill \
= {left( {1 times 3} right)^2} hfill \
= {3^2} hfill \
= 9 hfill \
hfill \
{text{h) }}{left[ {{{left( b right)}^{ – a}}} right]^{ – c}} = {left[ {{{left( 3 right)}^{ – 1}}} right]^{ – 2}} hfill \
= {3^{left( { – 1} right) times left( { – 2} right)}} = {3^2=9} hfill \
end{gathered} ]
{text{In this excercise we shall substitute the following numbers}} hfill \
a = 1,{text{ }}b = 3,{text{ }}c = 2 hfill \
{text{a) }}a{c^c} = 1{left( 2 right)^2} hfill \
{text{Evaluate the power and then multiply}} hfill \
= 1left( 4 right) hfill \
= 4 hfill \
hfill \
{text{b) }}{a^c}{b^c} = {1^2}left( {{3^2}} right) hfill \
{text{Evaluate the power and then multiply}} hfill \
= 1left( 9 right) hfill \
= 9 hfill \
hfill \
{text{c) }}{left( {ab} right)^{ – c}} = {left[ {left( 1 right)left( 3 right)} right]^{ – 2}} hfill \
{text{Multiply and evaluate the power}} hfill \
= {left( 3 right)^{ – 2}} hfill \
{text{Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
= frac{1}{{{3^2}}} hfill \
= frac{1}{9} hfill \
hfill \
{text{d) }}{left( {b div c} right)^{ – a}} = {left( {3 div 2} right)^{ – 1}} hfill \
= {left( {frac{3}{2}} right)^{ – 1}} hfill \
{text{Remember this rule, }}{left( {frac{a}{b}} right)^{ – n}} = frac{1}{{{{left( {frac{a}{b}} right)}^n}}} = {left( {frac{b}{a}} right)^n} hfill \
= left( {frac{2}{3}} right) hfill \
hfill \
{text{e) }}{left( { – a div b} right)^{ – c}} = {left( { – 1 div 3} right)^{ – 2}} hfill \
= {left( {frac{{ – 1}}{3}} right)^{ – 2}} hfill \
{text{Remember this rule, }}{left( {frac{a}{b}} right)^{ – n}} = frac{1}{{{{left( {frac{a}{b}} right)}^n}}} = {left( {frac{b}{a}} right)^n} hfill \
= {left( {frac{3}{{ – 1}}} right)^2} = {left( { – 3} right)^2} hfill \
{text{Negative numbers raised to an even power is always positive}} hfill \
= 9 hfill \
hfill \
{text{f) }}{left( {{a^{ – 1}}{b^{ – 2}}} right)^c} = {left[ {left( {{1^{ – 1}}} right)left( {{3^{ – 2}}} right)} right]^2} hfill \
{text{Remember this rule, }}{a^{ – n}} = frac{1}{{{a^n}}} hfill \
= {left( {frac{1}{1} times frac{1}{{{3^2}}}} right)^2} hfill \
{text{Evaluate and simplify the expression}} hfill \
= {left( {1 times frac{1}{9}} right)^2} hfill \
= {left( {frac{1}{9}} right)^2} hfill \
= frac{1}{{81}} hfill \
hfill \
{text{g) }}{left( {{a^b}{b^a}} right)^c} = {left( {{1^3} times {3^1}} right)^2} hfill \
{text{Multiply and evaluate the powers}} hfill \
= {left( {1 times 3} right)^2} hfill \
= {3^2} hfill \
= 9 hfill \
hfill \
{text{h) }}{left[ {{{left( b right)}^{ – a}}} right]^{ – c}} = {left[ {{{left( 3 right)}^{ – 1}}} right]^{ – 2}} hfill \
= {3^{left( { – 1} right) times left( { – 2} right)}} = {3^2=9} hfill \
end{gathered} ]
Result
2 of 2
a) 4
b) 9
c) $dfrac{1}{9}$
d) $dfrac{2}{3}$
e) 9
f) $dfrac{1}{81}$
g) 9
h) 9
Exercise 15
Step 1
1 of 4
Remember the following rule for exponents:
$(-a)^n=(-1)^n cdot a^n$ where
$$
{left( { – 1} right)^n} = left{ {begin{array}{c}
{1{text{ if }}n{text{ is even}}} \
{ – 1{text{ if }}n{text{ is odd}}}
end{array}} right.
$$
$$
a^{-n}=dfrac{1}{a^n}
$$
Step 2
2 of 4
a) Evaluate each case. The exponent $3$ is odd so $(-a)^3=-a^3$
$(-10)^3=(-1)^3cdot 10^3=-1times 1000=-1000$
On the other hand $10^{-3}$ has negative exponent so it goes at the denominator.
$10^{-3}=dfrac{1}{10^3}=dfrac{1}{1000}=0.001$
Therefore, $(-10)^3$ is $(-10)$ multiplied by itself 3 times while $10^-3$ is the reciprocal of $10^3$
Step 3
3 of 4
b) The exponent $4$ is even, so $(-a)^4=(-1)^4cdot a^4$
$(-10)^4=(-1)^4cdot 10^4=10,000$
In $-10^4$, the negative sign is not enclosed in parenthesis so it becomes
$-10^4=-1times 10^4=-1times 10,000=-10,000$
Therefore, $(-10)^4$ is $(-10)$ multiplied by itself 4 times while $-10^4$ is the negative of $10^4$
Result
4 of 4
a) $(-10)^3$ is $(-10)$ multiplied by itself 3 times while $10^{-3}$ is the reciprocal of $10^3$
b) $(-10)^4$ is $(-10)$ multiplied by itself 4 times while $-10^4$ is the negative of $10^4$
Exercise 16
Step 1
1 of 2
[begin{gathered}
{text{a) Factor such that both sides have the same base}}{text{.}} hfill \
{16^x} = frac{1}{{16}}{text{ }} hfill \
{16^x} = {16^{ – 1}} hfill \
x{text{ = }} – 1 hfill \
hfill \
{text{b) Write the number in exponential form with a base of 10}} hfill \
{text{1}}{{text{0}}^x} = 0.01 hfill \
{10^x} = {10^{ – 2}} hfill \
x = – 2 hfill \
hfill \
{text{c) Remember this rule, }}{x^0} = 1 hfill \
{text{Therefore}} hfill \
{{text{2}}^x} = 1 hfill \
x = 0 hfill \
hfill \
{text{d) Write the number in exponential form with a base of 2}} hfill \
{2^n} = {0.5^2} hfill \
{2^n} = {left( {frac{1}{2}} right)^2} hfill \
{{text{2}}^n} = {2^{ – 2}} hfill \
n = – 2 hfill \
hfill \
{text{e) Factor such that both sides have the same base}}. hfill \
{text{2}}{{text{5}}^n} = frac{1}{{625}} hfill \
{left( {{5^2}} right)^n} = frac{1}{{{5^4}}} hfill \
{5^{2n}} = {5^{ – 4}} hfill \
2n = – 4 hfill \
{text{Divide both sides by 2}} hfill \
n = – 2 hfill \
hfill \
{text{f) Factor such that both sides have the same base}}. hfill \
{text{1}}{{text{2}}^n} = frac{1}{{144}} hfill \
{12^n} = {12^{ – 2}} hfill \
n = – 2 hfill \
end{gathered} ]
{text{a) Factor such that both sides have the same base}}{text{.}} hfill \
{16^x} = frac{1}{{16}}{text{ }} hfill \
{16^x} = {16^{ – 1}} hfill \
x{text{ = }} – 1 hfill \
hfill \
{text{b) Write the number in exponential form with a base of 10}} hfill \
{text{1}}{{text{0}}^x} = 0.01 hfill \
{10^x} = {10^{ – 2}} hfill \
x = – 2 hfill \
hfill \
{text{c) Remember this rule, }}{x^0} = 1 hfill \
{text{Therefore}} hfill \
{{text{2}}^x} = 1 hfill \
x = 0 hfill \
hfill \
{text{d) Write the number in exponential form with a base of 2}} hfill \
{2^n} = {0.5^2} hfill \
{2^n} = {left( {frac{1}{2}} right)^2} hfill \
{{text{2}}^n} = {2^{ – 2}} hfill \
n = – 2 hfill \
hfill \
{text{e) Factor such that both sides have the same base}}. hfill \
{text{2}}{{text{5}}^n} = frac{1}{{625}} hfill \
{left( {{5^2}} right)^n} = frac{1}{{{5^4}}} hfill \
{5^{2n}} = {5^{ – 4}} hfill \
2n = – 4 hfill \
{text{Divide both sides by 2}} hfill \
n = – 2 hfill \
hfill \
{text{f) Factor such that both sides have the same base}}. hfill \
{text{1}}{{text{2}}^n} = frac{1}{{144}} hfill \
{12^n} = {12^{ – 2}} hfill \
n = – 2 hfill \
end{gathered} ]
Result
2 of 2
a) $x=-1$
b) $x=-2$
c) $x=0$
d) $n=-2$
e) $n=-2$
f) $w=-2$
Exercise 17
Step 1
1 of 2
Remember this rule of exponent:
$(a^m)^n=a^{mtimes n}$
$a^{-n}=dfrac{1}{a^n}$
In this case, we are given that
$10^{2y}=25$
To find $10^{-y}$, raise both sides to the power of $-dfrac{1}{2}$
$left(10^{2y}right)^{-1/2}=25^{-1/2}$
$(10^{-y})=25^{-1/2}$
$10^{-y}=(5^2)^{-1/2}$
$10^{-y}=5^{-1}$
$$
10^{-y}=dfrac{1}{5}
$$
Result
2 of 2
$$
10^{-y}=dfrac{1}{5}
$$
10^{-y}=dfrac{1}{5}
$$
Exercise 18
Step 1
1 of 2
[begin{gathered}
{text{a) Remember this rule, (}}{a^n}{)^m} = {a^{n cdot m}} hfill \
{left( {{x^2}} right)^{5 – r}} hfill \
= {x^{2left( {5 – r} right)}} hfill \
= {x^{10 – 2r}} hfill \
hfill \
{text{b) Remember this rule, }}{{text{a}}^n} div {a^m} = {a^{n – m}} hfill \
left( {{b^{2m + 3m}}} right) div left( {{b^{m – n}}} right) hfill \
= left( {{b^{5m}}} right) div left( {{b^{m – n}}} right) hfill \
= {b^{5m – left( {m – n} right)}} hfill \
= {b^{4m + n}} hfill \
hfill \
{text{c) Remember this rule, }}{{text{a}}^n} div {a^m} = {a^{n – m}} hfill \
left( {{b^{2m + 3n}}} right) div left( {{b^{m – n}}} right) hfill \
= {b^{2m + 3n – left( {m – n} right)}} hfill \
= {b^{m + 4n}} hfill \
hfill \
{text{d) Remember this rule, }}{a^n}left( {{a^m}} right) = {a^{n + m}}{text{ }} hfill \
{{text{x}}^{3left( {7 – r} right)}}{x^r} hfill \
= {x^{3left( {7 – r} right) + r}} hfill \
= {x^{21 – 3r + r}} hfill \
= {x^{21 – 2r}} hfill \
hfill \
{text{e) Remember this rule, }}{{text{a}}^n} div {a^m} = {a^{n – m}} hfill \
{text{(}}{{text{a}}^{10 – p}}){left( {frac{1}{a}} right)^p} hfill \
= {text{(}}{{text{a}}^{10 – p}})left( {frac{1}{{{a^p}}}} right) hfill \
= {a^{10 – p – p}} hfill \
= {a^{10 – 2p}} hfill \
hfill \
{text{f) Remember this rule, (}}{a^n}{)^m} = {a^{n cdot m}} hfill \
left[ {{{left( {3{x^4}} right)}^{6 – m}}} right]{left( {frac{1}{x}} right)^m} hfill \
= left( {{3^{1left( {6 – m} right)}}{x^{4left( {6 – m} right)}}} right)left( {frac{1}{{{x^m}}}} right) hfill \
{text{Remember this rule, }}{{text{a}}^n} div {a^m} = {a^{n – m}} hfill \
= left( {{3^{6 – m}}{x^{24 – 4m}}} right)left( {frac{1}{{{x^m}}}} right) hfill \
= {3^{6 – m}}{x^{24 – 4m – m}} hfill \
= {3^{6 – m}}{x^{24 – 5m}} hfill \
end{gathered} ]
{text{a) Remember this rule, (}}{a^n}{)^m} = {a^{n cdot m}} hfill \
{left( {{x^2}} right)^{5 – r}} hfill \
= {x^{2left( {5 – r} right)}} hfill \
= {x^{10 – 2r}} hfill \
hfill \
{text{b) Remember this rule, }}{{text{a}}^n} div {a^m} = {a^{n – m}} hfill \
left( {{b^{2m + 3m}}} right) div left( {{b^{m – n}}} right) hfill \
= left( {{b^{5m}}} right) div left( {{b^{m – n}}} right) hfill \
= {b^{5m – left( {m – n} right)}} hfill \
= {b^{4m + n}} hfill \
hfill \
{text{c) Remember this rule, }}{{text{a}}^n} div {a^m} = {a^{n – m}} hfill \
left( {{b^{2m + 3n}}} right) div left( {{b^{m – n}}} right) hfill \
= {b^{2m + 3n – left( {m – n} right)}} hfill \
= {b^{m + 4n}} hfill \
hfill \
{text{d) Remember this rule, }}{a^n}left( {{a^m}} right) = {a^{n + m}}{text{ }} hfill \
{{text{x}}^{3left( {7 – r} right)}}{x^r} hfill \
= {x^{3left( {7 – r} right) + r}} hfill \
= {x^{21 – 3r + r}} hfill \
= {x^{21 – 2r}} hfill \
hfill \
{text{e) Remember this rule, }}{{text{a}}^n} div {a^m} = {a^{n – m}} hfill \
{text{(}}{{text{a}}^{10 – p}}){left( {frac{1}{a}} right)^p} hfill \
= {text{(}}{{text{a}}^{10 – p}})left( {frac{1}{{{a^p}}}} right) hfill \
= {a^{10 – p – p}} hfill \
= {a^{10 – 2p}} hfill \
hfill \
{text{f) Remember this rule, (}}{a^n}{)^m} = {a^{n cdot m}} hfill \
left[ {{{left( {3{x^4}} right)}^{6 – m}}} right]{left( {frac{1}{x}} right)^m} hfill \
= left( {{3^{1left( {6 – m} right)}}{x^{4left( {6 – m} right)}}} right)left( {frac{1}{{{x^m}}}} right) hfill \
{text{Remember this rule, }}{{text{a}}^n} div {a^m} = {a^{n – m}} hfill \
= left( {{3^{6 – m}}{x^{24 – 4m}}} right)left( {frac{1}{{{x^m}}}} right) hfill \
= {3^{6 – m}}{x^{24 – 4m – m}} hfill \
= {3^{6 – m}}{x^{24 – 5m}} hfill \
end{gathered} ]
Result
2 of 2
a) $x^{10-2r}$
b) $b^{4m-n}$
c) $b^{m+2n}$
d) $x^{21-2r}$
e) $a^{10-2p}$
f) $3^{6-m}x^{24-5m}$
unlock