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(1a2) Add Fractions

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1) Convert to common denominator 2) Add the numerators 3) Reduce the fraction

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(1a3) Find opposite of a fraction

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1) Switch numerator and denominator 2) Change sign

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(1a4) Subtract two negative numbers

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-1 -2 = -3

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(1a5) Subtract 1 - (-2)

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1 - (-2) = 1 + 2

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(1a6) Subtract Fractions

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1) Convert to common denominator 2) Subtract the numerators 3) Reduce the fraction

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(1a7) Multiply negatives

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1) Two negatives = negative result 2) One negative, One positive = positive result

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(1a8) Multiply fractions

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1 ) Multiply the numerators 2) Multiply the denominators

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(1a9) Reciprocal of a fraction

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1) Switch numerator and denominator 2) Do not change sign

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Opposite vs. Reciprocal of a fraction

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OPPOSITE 1) Switch numerator and denominator 2) Change sign RECIPROCAL 1) Switch numerator and denominator 2) Do not change sign

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(1a10) Divide Fractions

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1) Switch numerator and denominator of second fraction 2) Multiply numerators and denominators

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(1a10) Divide Negatives

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1) -/- = + 2) - /+ or +/- = -

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(1b1 and 1b2) Exponents

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1) Multiply the base number by itself, the number of times that the exponent says 3 to the third (3^3)= 3x3x3

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(1b3 and 1b4) A negative number with an exponent

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1) Raise a negative to an even power = positive result 2) Raise a negative to an odd power = negative result

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(1b5) A fraction with an exponent

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1) Multiply the fraction by itself, the number of times that the exponent says. 2) Reduce the fraction

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(1b6) What is a square root?

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1) The square root of a number is the number that when multiplied by itself produces the number given.

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(1b7) The square root of a fraction

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1) The answer is in the form of a fraction

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(1b8 and 1b9) The little number on top of the square root symbol

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1) It's called the "index" 2) It means that instead of the "square root of a" you are now considering the "nth root of a". Eg: The square (2) root of 9 = 3 because 3 x 3= 9 The cubed (3) root of 8 = 2 because 2 x 2 x 2 = 8 The fourth (4) root of 16 = 2 because 2 x 2 x 2 x 2 = 16

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Why are "roots" (square roots) or "radicals" (synonym) the "opposite" of exponents or "powers"?

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A root can "undo" an exponent. An exponent can "undo" a radical Eg: 2 squared = 4 Square root of 4 = 2

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How can you rewrite the cubed root of 216 so that it is expressed as an exponent?

answer

216 with an exponent of 1/3

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(1c 1 - 10) Order of operations

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PEMDAS "Please Excuse My Dear Aunt Sally" 1) Parentheses (or any kind of bracket) 2) Exponents (or square roots) 3) Multiplication and Division (in order from left to right) 4) Addition and Subtraction (in order from left to right)

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(2-1) What does it mean when letters are scrunched together with no signs in between them?

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Multiply them.

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(2-2) What do the little numbers down and to the right of big numbers mean?

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Different things in different situations. In graphing you might get x1 and x2 just to show you changed something and are comparing. In algebra you may practice doing math with x1, x2, y1, y2 but it just means it's a different variable.

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(2-3) Combine like terms

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2y + 2y = 4y

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(2-4) What if a letter is squared?

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Then you can have that squared letter be part of the answer.

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(2-5) What if you can't combine like terms any more to get an answer?

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The answer can have more than one term like 9p+18 You just simplify it as much as you can.

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(2-6) Can you combine 10x with 16x squared?

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No. These are separate terms that might both be in your answer.

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(2-7;8) Distribute: 3(5x-2)

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3(5x-2) = 15x-6 (3 and 5x can be multiplied even though you don't know x because 3*5*x will come out the same as 3*5x)

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(2-8) Polynomial

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An expression of more than two algebraic terms

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(2-8) Polynomial in standard form

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Order according to size of exponents, from biggest to smallest: 2x(cubed) - 7x(squared) + 2x -4

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(2-9) Multiply variables with exponents

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Add the exponents and combine the terms y^2 * y^2 = y^4 Because 2 + 2 = 4

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(2-10) Divide variables with exponents

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Subtract the exponents and combine terms y(to the fifth) / y(squared) = y(cubed) because 5 - 2 = 3

question

Add or subtract variables with exponents

answer

Trick question. You can't combine because they are not like terms.

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(2-12) Raise a power to a power. (y^3)^2

answer

Multiply the exponents.

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(2-13) The zero exponent rule

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Anything to a power of zero equals 1. See graphic for why

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(2-14 ; 16) Express a term with a negative exponent as an equivalent term with a positive exponent.

answer

9^-3 = 1/9^3 See graphic for why

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(2-15) A negative number raised to the power of a negative number.

answer

1) Get rid of the negative on the base number by rewriting the problem as -1* y to the negative power 2) Get rid of the negative on the exponent by rewriting the problem as -1 * 1/y to the positive power 3) Solve the fraction 4) Multiply the whole fraction by negative 1 which just puts a negative sign in front of it.

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(2-17) -2t^-5

answer

1) Rewrite as -2 x t ^-5 2)Convert the t^-5 to 1/t^5 3) Multiply -2 by that fraction, which puts the -2 in the numerator. That's as simplified as it can get. That's your answer.

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(2-18) Rewrite 1/y^-5 using only positive exponents.

answer

y^5 (Reverse a process learned earlier)

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(2-19) The Product Rule

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When multiplying exponential expression with the same base, add the exponents. Use this sum as the exponent of the common base.

question

(2-20) The Quotient Rule

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When dividing exponential expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator. Use this difference as the exponent of the common base.

question

(2-21) If an exponent turns out to be a 1

answer

Write the variable or base without an exponent

question

(2-21) If an exponent turns out to be a zero

answer

Omit that variable or base from the solution

question

(2-21) Many numbers, letters, and exponents on both sides of a fraction. (Together to be multiplied)

answer

1) Separate it out into separate fractions to be multiplied, if the base can be the same on top and bottom of each fraction. 2) Simplify each fraction. (Subtract exponents from each other so you end up with only one of each base variable.) 3) Convert any negative exponents into positive ones. 4) Multiply them all together. (3 ; 4 mean that positive exponents go on the top and negative exponents go on bottom as positive exponents.)

question

(2-22) A number with a fraction for an exponent

answer

1) Rewrite as a square root of the base, with the denominator of the fractional exponent as the index (the little number above the square root symbol). (This assumes that the fraction has a 1 for the denominator.) 2) Find the square root (or cubed root, etc)

question

(2-23) A negative base with a fractional exponent

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1) Get rid of the negative sign (temporarily) by rewriting the problem as -1* the rest 2) Convert the fractional exponent into a root problem and find the root. (1/2 exponent = square root) 3) Multiply the result by -1.

question

(2-24) x * x squared

answer

x cubed

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(2-24) x * 8

answer

8x

question

(2-25) Coefficient

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a numerical or constant quantity placed before and multiplying the variable in an algebraic expression

question

(2-25) Multiply: -2y^2 * 2y^5

answer

1) Multiply the constants together 2) Multiply the y's together by adding the exponents 3) Put the constant back together with the y value

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(2-25) Multiply: -2y^2 * 7

answer

-14y^2

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(2-26) Multiply (y-1)(a bunch of stuff)

answer

Rewrite as (y)(stuff) + (-1)(stuff)

question

(2-27 & 30) The FOIL method

answer

First Outside Inside Last For multiplying two groups of two terms together. It can be used to turn the expression into a polynomial, commonly a trinomial because you can combine terms.

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(2-28) The Difference of Two Squares Formula

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(x+a)(x-a)= x^2 - a^2

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(2-29) The Special Product Formula

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(x-a)^2 = x^2 - 2ax + a^2

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(2-31) Factor out (or write in a factored form) 10x-5

answer

5(2x-1)

question

(2-32 & 33) Factor out the GCF: 3y^6 - 15xy^7

answer

1) Find the GCF of the coefficients 2) Find the GCF of the variables 3) Multiplied together, they are the GCF of the polynominal

question

(2-34) What do you do when you are factoring out the GCF and the GCF happens to be one of the terms?

answer

Replace that term with a 1, since 1 x that term is that term

question

(2-34) Factoring out is the opposite of what, and why?

answer

The Distributive Property a(b+c)=ab+ac Factor ab+ac = a(b+c)

question

(2-34) Detailed illustration of factoring out the GCF

answer

4x^2 - 6x^3 + 2x 2*2*x*x - 2*3*x*x*x + 2*x Common Factor: 2x What's left? 2x - 3x^2 + 1 Solution: 2x(2x-3x^2+1)

question

(2-34 ; 35) Factor out the CGF when the GCF is a binomial

answer

8x(z+16) + (z+16) (z+16)( ? ) (z+ 16)(8x + 1)

question

(2-36 ; 37) What can you do if you are asked to factor out the GCF of a polynomial, but you can't see any common factors?

answer

1) Group it into two binomials 2) factor out the GCF of each group. 3) This may yield a binomial that is a common factor of the resulting two terms.

question

(2-38) Factor a trinomial

answer

1) Arrange in standard format (Descending order of exponents) 2) Factor out any GCF

question

(2-38) What happens if there is no common factor in a trinomial?

answer

Use a reverse FOIL method to convert the trinomial into two binomials. (IF the trinomial is in the form x^2+bx+c)

question

(2-38 - 42) Factor a trinomial of the form x^2 + bx + c or ax^2 + bx + c

answer

The FOIL method usually converts two binomials into a trinomial. Now we're turning a trinomial (ax^2 +bx +c) into two binomials. So this functions as a reverse FOIL method. (The course material taught one method for when a = 1 and another method for when a = not 0 or 1. I like to use the "harder" method for both so I don't get confused. I've re-formatted the method the way it makes sense to me. A memory tool for my method's acronym: Yoda, frustrated with all this algebra work, says, "A flippin' reward, gotta see.") 1) ARRANGE the trinomial into descending order if necessary. 2) FIND a set of numbers that when multiplied = ac and when added = b 2-a) Draw: blank * blank = ac blank + blank = b 2-b) Determine sign first if helpful. Positive result and positive sum: ++ Positive result and negative sum: - - Negative result and negative sum: + - 3) REWRITE the trinomial only this time make the middle term two terms by replacing b with the two numbers you found. 4) GROUP into two binomials and factor them out individually 5) SEE the new common factor (a binomial) and make the remaining factors as its own binomial.

question

(2-43-45) What is one way that you might be able to simplify a fraction with complex algebraic expressions as the numerator and the denominator?

answer

1) Factor out the numerator and denominator 2) Divide out any factors that are common between the numerator and the denominator.

question

(2-46) Perform the indicated operation and simplify the result. Leave your answer in factored form. 11/x + 3/x

answer

1) If the denominator is the same you can just add the numerators together. 2) A fraction is apparently a "factored form" so don't be scared by this instruction. The answer is 14/x.

question

(2-47) Steps to simplify the addition of two fractions with algebraic terms in the numerators and denominators.

answer

1) If the denominators are the same, add the numerators. 2) Combine like terms. 3) Simplify by factoring out the numerator and denominator if possible. 4) Simplify by dividing out terms common to the numerator and denominator if possible.

question

(2-48 & 49) Add and Subtract fractions that have algebraic expressions as the numerator and the denominator, but do not have common denominators.

answer

1) Find the lowest common denominator. 1-a Try factoring the denominators by using exponents to make them more similar. 1-b After they are in their simplest factored forms, you can find the LCD of exponential expressions by listing all the common factors to their highest power, and all the non-common factors, then simplify again. 2-c Try multiplying them together if that's all you can do. 2) Rewrite the factions using the LCD you found, multiplying the numerators by the appropriate amount. (The amount by which you multiplied the denominator to turn it into the LCD.) 3) Perform the addition or subtraction as required. 4) Factor out the numerator. 5) Divide out anything common to the numerator and the denominator.

question

(2-49) The process of simplification sometimes uses the factoring and distributive operations both ways, multiple times. Rank the following in order from most complex to simplest. 5(x+9) - 3(x+1) 5x + 45 - 3x + 3 2x+48 2(x+24)

answer

They are in order.

question

(2-49) Which is more "simplified", 2x+48 or 2(x+24)?

answer

2(x+24)

question

(2-49) Which is more "simplified", 5(x+9) - 3(x+1) or 5x + 45 - 3x + 3

answer

5x + 45 - 3x + 3

question

(2-49) Which is more "simplified", 5x + 45 - 3x + 3 or 2x + 48

answer

2x + 48

question

(2-49) Which is more "simplified", 5(x+9) - 3(x+1) or 2(x+24)

answer

2(x+24)

question

(3) Rational Expression

answer

A rational equation is an equation in which one or more of the terms is a fractional one. When solving these rational equations, we utilize one of two methods that will eliminate the denominator of each of the terms. (Got this from a website. The two methods are cross-multiplication (or flipping the second fraction and multiplying across), and changing the denominator to a common denominator. Not clear yet how these methods eliminate the denominator.)

question

(3-1) Solve linear equations

answer

1. To eliminate parentheses, use the distributive property where applicable. 2. Isolate the variable that you are solving for, by moving things from one side to the other. 3. Move things from one side to the other by doing the same operation to both sides, choosing an operation that will cancel out an expression on one side. For example, move 3 to the other side by adding negative 3 to both sides. Multiplying and Dividing are opposites like adding and subtracting are. 4. Once you solve for the variable, check it by plugging it into the original equation.

question

(3-2&3) What do you do with this? -(x-7)

answer

Distribute a negative 1

question

(3-4) How do you clear fractions from an equation?

answer

1. Multiply both sides of the equation by the LCD of all the fractions in the equation. Put a parentheses around the whole side of the equation and multiply that by the LCD. 2. Use the distributive property to multiply the LCD by each of the terms in the equation in turn. 3.

question

(3-4) How do you multiply a number by a fraction?

answer

1. Any number is equal to that number over 1, so make a fraction out of the number. 2. Multiply the fractions by multiplying the numerators and multiplying the denominators. 3. Reduce the fraction if possible

question

(3-4) What is a short cut for solving this? 6/1 * 1/3

answer

6/3 = 2 Long Way: 6/1 * 1/3 = 6/3 = 2/1 = 2

question

(3-4) What is a shortcut for solving this? 6/1 * (x+1)/2

answer

6/2 = 3 so you're left with 3(x+1) Long Way: 6/1 * (x+1)/2 = 6(x+1)/2 = 6/2(x+1) = 3(x+1)

WGU College Algebra (In Progress – based on homework)

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