All Solutions
Page 239: Practice Questions
$$
x^{a}x^{b}=x^{a+b}
$$
dfrac{x^a}{x^b}=x^{a-b}
$$
(x^a)^b=x^{ab}
$$
$$
=3^{7-5}=3^2
$$
$$
dfrac{x^a}{x^b}=x^{a-b}
$$
$=left(dfrac{1}{10}right)^{6-4}$
$=left(dfrac{1}{10}right)^2$
$$
dfrac{x^a}{x^b}=x^{a-b}
$$
$=left(7^{2-4}right)^{-1}$
$=left(7^{-2}right)^{-1}$
$=left(7^{-2(-1)}right)$
$=7^2$
$$
(x^a)^b=x^{ab}
$$
b) $-dfrac{1}{8}$
c) $9^{18}$
d) $3^2$
e) $left(dfrac{1}{10}right)^2$
f) $7^2$
a),,{4^{ – 2}} – {8^{ – 1}} hfill \
= frac{1}{{{4^2}}} – frac{1}{8} hfill \
= frac{1}{{16}} – frac{1}{8} hfill \
= frac{1}{{16}} – frac{{1left( 2 right)}}{{8left( 2 right)}} hfill \
= frac{1}{{16}} – frac{2}{{16}} hfill \
= -frac{{ 1}}{{16}} hfill \
end{gathered} ]
$$
x^{-a}=dfrac{1}{x^a}
$$
{text{b) }}{left( {4 + 8} right)^0} – {5^{ – 2}} hfill \
= {left( {12} right)^0} – {5^{ – 2}} hfill \
= 1 – {5^{ – 2}} hfill \
= 1 – frac{1}{{{5^2}}} hfill \
= 1 – frac{1}{{25}} hfill \
= frac{{25}}{{25}} – frac{1}{{25}} hfill \
= frac{{24}}{{25}} hfill \
end{gathered} ]
$x^0=1$ for all $x$ except $x=0$
$$
x^{-a}=dfrac{1}{x^a}
$$
{text{c) 2}}{{text{5}}^{ – 1}} + 3{left( {{5^{ – 1}}} right)^2} hfill \
= {25^{ – 1}} + 3left( {{5^{ – 2}}} right) hfill \
= frac{1}{{25}} + frac{3}{{{5^2}}} hfill \
= frac{1}{{25}} + frac{3}{{25}} hfill \
= frac{4}{{25}} hfill \
end{gathered} ]
$(x^a)^b=x^{ab}$
$$
x^{-a}=dfrac{1}{x^a}
$$
$=(-1)^3cdot left(dfrac{1}{2}right)^3+4^{-3}$
$=-1cdot dfrac{1}{2^3}+4^{-3}$
$=-1cdot dfrac{1}{8}+dfrac{1}{4^3}$
$=-dfrac{1}{8}+dfrac{1}{64}$
$=-dfrac{1(8)}{8(8)}+dfrac{1}{64}$
$=-dfrac{8}{64}+dfrac{1}{64}$
$$
=-dfrac{7}{64}
$$
$(-x)^a=(-1)^acdot x^a$
if $a$ is even, $(-1)^a=1$
if $a$ is odd, $(-1)^a=-1$
$$
x^{-a}=dfrac{1}{x^a}
$$
b) $dfrac{24}{25}$
c) $dfrac{4}{25}$
d) $-dfrac{7}{64}$
{text{a) }}{left( {frac{4}{7}} right)^2} = frac{{{4^2}}}{{{7^2}}} = frac{{16}}{{49}}
$$
$$
{left( {dfrac{x}{y}} right)^a} = dfrac{{{x^a}}}{{{y^a}}}
$$
{text{b) }}{left( { – frac{2}{5}} right)^3} = {left( { – 1} right)^3}{left( {frac{2}{5}} right)^3} = – 1 cdot frac{{{2^3}}}{{{5^3}}} = – frac{8}{{125}}
$$
{text{Use the following rules:}} hfill \
{left( { – x} right)^a} = {left( { – 1} right)^a}{x^a} hfill \
{left( { – 1} right)^a} = left{ {begin{array}{*{20}{c}}
{1{text{ if }}a{text{ is even}}} \
{ – 1{text{ if }}a{text{ is odd}}}
end{array}} right. hfill \
hfill \
{left( {frac{x}{y}} right)^a} = frac{{{x^a}}}{{{y^a}}} hfill \
end{gathered} ]
{text{c) }}{left( { – frac{2}{3}} right)^{ – 3}} = {left( { – frac{3}{2}} right)^3} = {left( { – 1} right)^3} cdot frac{{{3^3}}}{{{2^3}}} = – frac{{27}}{8}
$$
{text{Use the following rules:}} hfill \
{left( { – x} right)^a} = {left( { – 1} right)^a}{x^a} hfill \
{left( { – 1} right)^a} = left{ {begin{array}{*{20}{c}}
{1{text{ if }}a{text{ is even}}} \
{ – 1{text{ if }}a{text{ is odd}}}
end{array}} right. hfill \
hfill \
{left( {frac{x}{y}} right)^{ – a}} = {left( {frac{y}{x}} right)^a} = frac{{{y^a}}}{{{x^a}}} hfill \
end{gathered} ]
{text{d) }}frac{{{{left( { – 3} right)}^{ – 2}}}}{{{{left( { – 3} right)}^{ – 5}}}} = {left( { – 3} right)^{ – 2 – left( { – 5} right)}} = {left( { – 3} right)^{ – 2 + 5}} hfill \
= {left( { – 3} right)^3} = {left( { – 1} right)^3} cdot {3^3} = – 27 hfill \
hfill \
end{gathered} ]
{text{Use the following rules:}} hfill \
{left( { – x} right)^a} = {left( { – 1} right)^a}{x^a} hfill \
{left( { – 1} right)^a} = left{ {begin{array}{*{20}{c}}
{1{text{ if }}a{text{ is even}}} \
{ – 1{text{ if }}a{text{ is odd}}}
end{array}} right. hfill \
hfill \
frac{{{x^a}}}{{{x^b}}} = {x^{a – b}} hfill \
end{gathered} ]
b) $-dfrac{8}{125}$
c) $-dfrac{27}{8}$
d) $-27$
{text{ We know that }}{x^{frac{1}{2}}} = sqrt x {text{ and }}x{text{ is restricted}} hfill \
{text{to non-negative values, that is, }}x geq 0 hfill \
{text{For }}{x^{ – frac{1}{2}}} = frac{1}{{sqrt x }},{text{ the denominator can NOT be }} hfill \
{text{zero so it is restricted to }}x > 0 hfill \
hfill \
{text{Thus, }}{x^{ – frac{1}{2}}}{text{ and }}{x^{frac{1}{2}}}{text{ have different restrictions}}{text{.}} hfill \
hfill \
hfill \
end{gathered} ]
{x^{ – frac{1}{2}}}{text{ is restricted to }}x > 0 hfill \
{x^{frac{1}{2}}}{text{ is restricted to }}x geq 0 hfill \
end{gathered} ]
{text{a) }}{left( {frac{{49}}{{81}}} right)^{frac{1}{2}}} = sqrt {frac{{49}}{{81}}} = frac{{sqrt {49} }}{{sqrt {81} }} = frac{{sqrt {{7^2}} }}{{sqrt {{9^2}} }} = frac{7}{9}
$$
$x^{frac{n}{m}}=sqrt[m]{x^n}$
$sqrt{dfrac{a}{b}}=dfrac{sqrt{a}}{sqrt{b}}$
$$
sqrt{a^2}=a
$$
{text{b) }}sqrt {frac{{100}}{{121}}} = frac{{sqrt {100} }}{{sqrt {121} }} = frac{{sqrt {{{10}^2}} }}{{sqrt {{{11}^2}} }} = frac{{10}}{{11}}
$$
$$
sqrt{a^2}=a
$$
{text{c) }}{left( {frac{{16}}{9}} right)^{ – 0.5}} = {left( {frac{9}{{16}}} right)^{0.5}} = {left( {frac{{{3^2}}}{{{4^2}}}} right)^{0.5}} = frac{3}{4}
$$
$$
(x^a)^b=x^{ab}
$$
{text{d) }}{left[ {{{left( { – 125} right)}^{frac{1}{3}}}} right]^{ – 3}} = {left( { – 125} right)^{ – 1}} = frac{1}{{left( { – 125} right)}} = – frac{1}{{125}}
$$
$$
x^{-a}=dfrac{1}{x^a}
$$
{text{e) }}sqrt[4]{{{{left( { – 9} right)}^{ – 2}}}} = sqrt[4]{{frac{1}{{{{left( { – 9} right)}^2}}}}} = sqrt[4]{{frac{1}{{81}}}} = {left( {frac{1}{{81}}} right)^{frac{1}{4}}} = {left( {frac{1}{{{3^4}}}} right)^{frac{1}{4}}} = frac{1}{3}
$$
$x^{-a}=dfrac{1}{x^a}$
$x^{frac{n}{m}}$ is undefined when
$x<0$ and $m$ is even number.
{text{f) }}frac{{ – sqrt[3]{{512}}}}{{sqrt[5]{{ – 1024}}}} = frac{{ – {{512}^{frac{1}{3}}}}}{{ – {{1024}^{frac{1}{5}}}}} = frac{{ – {{left( {{2^9}} right)}^{frac{1}{3}}}}}{{ – {{left( {{2^{10}}} right)}^{left( {frac{1}{5}} right)}}}} = frac{{ – {2^3}}}{{ – {2^2}}} = frac{{ – 8}}{{ – 4}} = 2
$$
$(x^a)^b=x^{ab}$
$x^{frac{n}{m}}$ is undefined when
$x<0$ and $m$ is even number.
b) $dfrac{10}{11}$
c) $dfrac{3}{4}$
d) $-dfrac{1}{125}$
e) $dfrac{1}{3}$
f) 2
$sqrt[m]{a^n}=a^{frac{n}{m}}$
To evaluate the expression, remember that
$$
(a^m)^{frac{1}{m}}=a
$$
defarraystretch{1.75}%
begin{tabular}{|l|l|l|}
hline
begin{tabular}[c]{@{}l@{}}Exponential\ Formend{tabular} & Radical Form & begin{tabular}[c]{@{}l@{}}Evaluation of\ Expressionend{tabular} \ hline
a) 100$^frac{1}{2}$ & $sqrt{100}$ & 10 \ hline
b) $16^{0.25}=16^{frac{1}{4}}$ & $sqrt[4]{16}$ & $sqrt[4]{2^4}=2$ \ hline
c) $ 121^{frac{1}{2}}$ & $sqrt{121}$ & $sqrt{11^2}=11$ \ hline
d) $ (-27)^{frac{5}{3}}$ & $sqrt[3]{(-27)^5}$ & $(-27)^{frac{5}{3}}=(-3^3)^{frac{5}{3}}=-3^5=-243$ \ hline
e) $ 49^{2.5}=49^{frac{5}{2}}$ & $sqrt[2]{49^5}$ & $49^{frac{5}{2}}=(7^2)^{frac{5}{2}}=7^5=16807$ \ hline
f) $ 1024^{frac{1}{10}}$ & $sqrt[10]{1024}$ & $1024^{frac{1}{10}}=(2^{10})^{frac{1}{10}}=2$ \ hline
end{tabular}
end{table}
defarraystretch{1.5}%
begin{tabular}{|l|l|l|}
hline
begin{tabular}[c]{@{}l@{}}Exponential\ Formend{tabular} & Radical Form & begin{tabular}[c]{@{}l@{}}Evaluation of\ Expressionend{tabular} \ hline
a) 100$^frac{1}{2}$ & $sqrt{100}$ & 10 \ hline
b) $16^{0.25}$ & $sqrt[4]{16}$ & $2$ \ hline
c) $121^{frac{1}{2}}$ & $sqrt{121}$ & $11$ \ hline
d) $(-27)^{frac{5}{3}}$ & $sqrt[3]{(-27)^5}$ & $-243$ \ hline
e) $49^{2.5}$ & $sqrt[2]{49^5}$ & $16807$ \ hline
f) $1024^{frac{1}{10}}$ & $sqrt[10]{1024}$ & $2$ \ hline
end{tabular}
end{table}
a) $-456^{frac{4}{7}}=-33.068$
b) $98^{0.25}=3.146$
c) $left(dfrac{5}{8}right)^{frac{5}{8}}=0.745$
d) $left(sqrt[5]{-1000}right)^3=(-1000)^{frac{3}{5}}=-63.096$
b) $31.147$
c) $0.745$
d) $-63.096$
$-8^{frac{4}{3}}=-(2^3)^{frac{4}{3}}=-(2^4)=-16$
$(-8)^{frac{4}{3}}=(-2^3)^{frac{4}{3}}=(-2)^4=16$
The difference is in how the negative sign is placed. The second one has negative sign inside the parenthesis while the first one does not. Remember that
$$
{left( { – x} right)^{frac{n}{m}}} = left{ {begin{array}{c}
{|x{|^{frac{n}{m}}}{text{ for odd }}m{text{ and even }}n} \
{ – |x{|^{frac{n}{m}}}{text{for odd }}m{text{ and odd }}n} \
{{text{undefined for even }}m}
end{array}} right.
$$
We can also see the difference when write in radical form.
$-8^{frac{4}{3}}=-sqrt[3]{8^4}$
$$
(-8)^{frac{4}{3}}=sqrt[3]{(-8)^4}
$$
$-8^{frac{4}{3}}=-sqrt[3]{8^4}=-16$
$$
(-8)^{frac{4}{3}}=sqrt[3]{(-8)^4}=16
$$
{text{a) }} hfill \
frac{{{x^{ – 3}}left( {{x^5}} right)}}{{{x^7}}} = frac{{{x^{ – 3 + 5}}}}{{{x^7}}} = frac{{{x^2}}}{{{x^7}}} = {x^{2 – 7}} = {x^{ – 5}} = frac{1}{{{x^5}}} hfill \
end{gathered} ]
$x^acdot x^b=x^{a+b}$
$dfrac{x^a}{x^b}=x^{a-b}$
$$
x^{-a}=dfrac{1}{x^a}
$$
{text{b) }} hfill \
frac{{{n^{ – 4}}left( {{n^{ – 6}}} right)}}{{{n^{ – 14}}}} = frac{{{n^{ – 4 + left( { – 6} right)}}}}{{{n^{ – 14}}}} = frac{{{n^{ – 10}}}}{{{n^{ – 14}}}} = {n^{ – 10 – left( { – 14} right)}} = {n^4} hfill \
end{gathered} ]
$x^acdot x^b=x^{a+b}$
$dfrac{x^a}{x^b}=x^{a-b}$
{text{c) }} hfill \
{left( {frac{{{{left( {{y^2}} right)}^6}}}{{{y^9}}}} right)^{ – 2}} = {left( {frac{{{y^{12}}}}{{{y^9}}}} right)^{ – 2}} = {left( {{y^{12 – 9}}} right)^{ – 2}} = {left( {{y^3}} right)^{ – 2}} = {y^{ – 6}} = frac{1}{{{y^6}}} hfill \
end{gathered} ]
$(x^a)^b=x^{ab}$
$x^acdot x^b=x^{a+b}$
$dfrac{x^a}{x^b}=x^{a-b}$
$$
x^{-a}=dfrac{1}{x^a}
$$
{text{d) }} hfill \
frac{{{{left( { – 2{x^5}} right)}^3}}}{{8{x^{10}}}} = frac{{{{( – 2)}^3}{x^{5(3)}}}}{{8{x^{10}}}} = frac{{-8{x^{15}}}}{{8{x^{10}}}} = {-x^{15 – 10}} = {-x^5} hfill \
end{gathered} ]
$(x^a)^b=x^{ab}$
$x^acdot x^b=x^{a+b}$
$dfrac{x^a}{x^b}=x^{a-b}$
{text{e) }} hfill \
{left( {3{a^2}} right)^{ – 3}} times {left( {9{a^{ – 1}}} right)^2} hfill \
= {3^{ – 3}}{a^{ – 6}} times {9^2}{a^{ – 2}} hfill \
= {3^{ – 3}}{a^{ – 6}} times 81{a^{ – 2}} hfill \
= {3^{ – 3}} times {a^{ – 6}} times {3^4}{a^{ – 2}} hfill \
= 3{a^{ – 6}}{a^{ – 2}} hfill \
{text{ = 3}}{{text{a}}^{ – 6 + left( { – 2} right)}} hfill \
= 3{a^{ – 8}} hfill \
{text{ = }}frac{3}{{{a^8}}} hfill \
end{gathered} ]
$(x^a)^b=x^{ab}$
$x^acdot x^b=x^{a+b}$
$$
x^{-a}=dfrac{1}{x^a}
$$
{text{f) }} hfill \
frac{{left( {4{r^{ – 6}}} right) times {{left( { – 2{r^2}} right)}^5}}}{{{{left( { – 2r} right)}^4}}} = frac{{4{r^{ – 6}} times {{left( { – 2} right)}^5}{r^{10}}}}{{{{left( { – 2r} right)}^4}}} hfill \
= frac{{4{r^{ – 6}} times left( { – 32} right){r^{10}}}}{{16{r^4}}} = frac{{ – 8{r^{10 – 6}}}}{{{r^4}}} hfill \
= – frac{{8{r^4}}}{{{r^4}}} = – 8 hfill \
end{gathered} ]
$(x^a)^b=x^{ab}$
$x^acdot x^b=x^{a+b}$
$dfrac{x^a}{x^b}=x^{a-b}$
b) $n^4$
c) $dfrac{1}{y^6}$
d) $-x^5$
e) $dfrac{3}{a^8}$
f) $-8$
{text{a) }} hfill \
frac{{{x^{0.5}}{y^{1.8}}}}{{{x^{0.3}}{y^{2.5}}}} = {x^{0.5 – 0.3}}{y^{1.8 – 2.5}} hfill \
= {x^{0.2}}{y^{ – 0.7}} hfill \
= frac{{{x^{0.2}}}}{{{y^{0.7}}}} hfill \
end{gathered} ]
$dfrac{x^a}{x^b}=x^{a-b}$
{text{b) }} hfill \
frac{{{{left( {m{n^3}} right)}^{ – frac{1}{2}}}}}{{{m^{frac{1}{2}}}{n^{ – frac{5}{2}}}}} = frac{{{m^{ – frac{1}{2}}}{n^{ – frac{3}{2}}}}}{{{m^{frac{1}{2}}}{n^{ – frac{5}{2}}}}} hfill \
hfill \
= {m^{ – frac{1}{2} – left( {frac{1}{2}} right)}}{n^{ – frac{3}{2} – left( { – frac{5}{2}} right)}} = {m^{ – 1}}n = frac{n}{m} hfill \
end{gathered} ]
$(x^a)^b=x^{ab}$
$dfrac{x^a}{x^b}=x^{a-b}$
$x^{-a}=dfrac{1}{x^a}$
{text{c)}} hfill \
frac{{sqrt {{x^2}{y^4}} }}{{{{left( {{x^{ – 2}}{y^3}} right)}^{ – 1}}}} = frac{{{{left( {{x^2}{y^4}} right)}^{frac{1}{2}}}}}{{{{left( {{x^{ – 2}}{y^3}} right)}^{ – 1}}}} = frac{{x{y^2}}}{{{x^2}{y^{ – 3}}}} = frac{{{y^5}}}{x} hfill \
end{gathered} ]
$(x^a)^b=x^{ab}$
$dfrac{x^a}{x^b}=x^{a-b}$
{text{d) }} hfill \
{left( {frac{{2ab{c^3}}}{{{{left( {2{a^2}{b^3}c} right)}^2}}}} right)^{ – 2}} = {left( {frac{{2ab{c^3}}}{{2{a^4}{b^6}{c^2}}}} right)^{ – 2}} hfill \
hfill \
= {left( {frac{c}{{{a^3}{b^5}}}} right)^{ – 2}} = {left( {frac{{2{a^3}{b^5}}}{c}} right)^2} hfill \
hfill \
= frac{{4{a^6}{b^{10}}}}{{{c^2}}} hfill \
end{gathered} ]
$left( x^a right)^b=x^{ab}$
$dfrac{x^a}{x^b}=x^{a-b}$
$x^{-a}=dfrac{1}{x^a}$
{text{e) }}frac{{sqrt[4]{{81{p^8}}}}}{{sqrt {9{p^4}} }} = frac{{{{left( {81{p^8}} right)}^{frac{1}{4}}}}}{{{{left( {9{p^4}} right)}^{frac{1}{2}}}}} = frac{{{{left( {{3^4}{p^8}} right)}^{frac{1}{4}}}}}{{{{left( {{3^2}{p^4}} right)}^{frac{1}{2}}}}} = frac{{3{p^2}}}{{3{p^2}}} = 1
$$
$sqrt[m]{x^n}=x^{frac{n}{m}}$
$(x^a)^b=x^{ab}$
$dfrac{x^a}{x^b}=x^{a-b}$
{text{f) }} hfill \
frac{{sqrt[6]{{{{left( {8{x^6}} right)}^2}}}}}{{sqrt[4]{{625{x^8}}}}} = frac{{{{left( {8{x^6}} right)}^{frac{2}{6}}}}}{{{{left( {625{x^8}} right)}^{frac{1}{4}}}}} = frac{{{{left( {{2^3}{x^6}} right)}^{frac{2}{6}}}}}{{{{left( {{5^4}{x^8}} right)}^{frac{1}{4}}}}} = frac{{{{left( {{2^3}{x^6}} right)}^{frac{1}{3}}}}}{{{{left( {{5^4}{x^8}} right)}^{frac{1}{4}}}}} hfill \
= frac{{{{left( {{2^3}} right)}^{frac{1}{3}}}{x^{6left( {frac{1}{3}} right)}}}}{{{{left( {{5^4}{x^8}} right)}^{frac{1}{4}}}}} = frac{{2{x^2}}}{{5{x^2}}} = frac{2}{5} hfill \
end{gathered} ]
$sqrt[m]{x^n}=x^{frac{n}{m}}$
$(x^a)^b=x^{ab}$
$dfrac{x^a}{x^b}=x^{a-b}$
b) $dfrac{n}{m}$
c) $dfrac{y^5}{x}$
d) $dfrac{4a^6b^{10}}{c^2}$
e) 1
f) $dfrac{2}{5}$
{text{a) }}{left( {frac{{{b^3}}}{{{a^{frac{5}{2}}}}}} right)^2}left( {frac{{2{a^4}}}{{{b^5}}}} right) = left( {frac{{{b^6}}}{{{a^5}}}} right)left( {frac{{2{a^4}}}{{{b^5}}}} right) = left( {frac{{{b^6}}}{{{a^5}}}} right)left( {frac{{2{a^4}}}{{{b^5}}}} right) hfill \
= 2{b^{6 – 5}}{a^{4 – 5}} = 2b{a^{ – 1}} = frac{{2b}}{a} hfill \
end{gathered} ]
$(x^a)^b=x^{ab}$
$x^acdot x^b=x^{a+b}$
$x^{-a}=dfrac{1}{x^a}$
{text{b) }}sqrt {frac{{9{b^3}{{left( {ab} right)}^2}}}{{{{left( {{a^2}{b^3}} right)}^3}}}} = sqrt {frac{{9{b^3}left( {{a^2}{b^2}} right)}}{{{{left( {{a^2}{b^3}} right)}^3}}}} = sqrt {frac{{9{a^2}{b^5}}}{{{a^6}{b^9}}}} hfill \
= {left( {frac{{9{a^2}{b^5}}}{{{a^6}{b^9}}}} right)^{frac{1}{2}}} = {left( {9{a^{2 – 6}}{b^{5 – 9}}} right)^{frac{1}{2}}} = {9^{frac{1}{2}}}{left( {{a^{ – 4}}{b^{ – 5}}} right)^{frac{1}{2}}} hfill \
= 3{a^{ – 2}}{b^{ – 5/2}} = frac{3}{{{a^2}{b^{frac{5}{2}}}}} hfill \
end{gathered} ]
$sqrt[m]{x^n}=x^{frac{n}{m}}$
$(x^a)^b=x^{ab}$
$$
x^{-a}=dfrac{1}{x^a}
$$
b) $dfrac{3}{a^2b^{frac{5}{2}}}$
{text{a) }}left( {{a^{10 + 2p}}} right)left( {{a^{ – p – 8}}} right) hfill \
= {a^{left( {10 + 2p} right) + left( { – p – 8} right)}} hfill \
= {a^{p + 2}} hfill \
end{gathered} ]
$(x^a)^b=x^{ab}$
$$
x^acdot x^b=x^{a+b}
$$
{text{b) }}{left( {2{x^2}} right)^{3 – 2m}}{left( {frac{1}{x}} right)^{2m}} hfill \
= {2^{3 – 2m}}{x^{2left( {3 – 2m} right)}}{x^{ – 2m}} hfill \
= {2^{3 – 2m}}{x^{2left( {3 – 2m} right) – 2m}} hfill \
= {2^{3 – 2m}}{x^{6 – 4m – 2m}} hfill \
= {2^{3 – 2m}}{x^{6 – 6m}} hfill \
end{gathered} ]
$(x^a)^b=x^{ab}$
$x^acdot x^b=x^{a+b}$
$$
left(dfrac{1}{x}right)^a=x^{-a}
$$
{text{c) }}left[ {{c^{2n – 3m}}} right]{left( {{c^3}} right)^m} div {left( {{c^2}} right)^n} hfill \
= {c^{2n – 3m}}{c^{3m}} div {c^{2n}} hfill \
= frac{{{c^{2n – 3m + 3m}}}}{{{c^{2n}}}} hfill \
= {c^{2n – 3m + 3m – 2n}} hfill \
= 1 hfill \
end{gathered} ]
$(x^a)^b=x^{ab}$
$x^acdot x^b=x^{a+b}$
$dfrac{x^a}{x^b}=x^{a-b}$
$$
left(dfrac{1}{x}right)^a=x^{-a}
$$
{text{d) }}left( {{x^{4n – m}}} right){left( {frac{1}{{{x^3}}}} right)^{m + n}} hfill \
= {x^{4n – m}} cdot {x^{ – 3left( {m + n} right)}} hfill \
= {x^{4n – m – 3left( {m + n} right)}} hfill \
= {x^{4n – m – 3m – 3n}} hfill \
= {x^{n – 4m}} hfill \
end{gathered} ]
$(x^a)^b=x^{ab}$
$x^acdot x^b=x^{a+b}$
$dfrac{x^a}{x^b}=x^{a-b}$
$$
left(dfrac{1}{x}right)^a=x^{-a}
$$
b) $2^{3-2m}x^{6-6m}$
c) $1$
d) $x^{n-4m}$
