WGU College Algebra (In Progress - based on homework)

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

Multiply the exponents.

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

Multiply them.

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

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

Opposite vs. Reciprocal of a fraction

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

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

Omit that variable or base from the solution

(2-8) Polynomial in standard form

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

(1c 1 - 10) Order of operations

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)

(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?

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

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

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

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

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

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

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.

(2-27 & 30) The FOIL method

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.

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

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

(2-20) The Quotient Rule

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.

(2-19) The Product Rule

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

(2-24) x * x squared

x cubed

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

y^5 (Reverse a process learned earlier)

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

-14y^2

(1a8) Multiply fractions

1 ) Multiply the numerators 2) Multiply the denominators

(1a5) Subtract 1 - (-2)

1 - (-2) = 1 + 2

(1a10) Divide Negatives

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

(2-38) Factor a trinomial

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

(1a2) Add Fractions

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

(1a6) Subtract Fractions

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

(2-9) Multiply variables with exponents

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

(2-8) Polynomial

An expression of more than two algebraic terms

(2-13) The zero exponent rule

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

(1a4) Subtract two negative numbers

-1 -2 = -3

(2-28) The Difference of Two Squares Formula

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

(2-29) The Special Product Formula

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

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

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.

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

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.

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

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.

(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?

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.

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

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.

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

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.

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

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

(1b1 and 1b2) Exponents

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

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

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

(1b5) A fraction with an exponent

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

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

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

(2-17) -2t^-5

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.

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

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)

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

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.)

(1a3) Find opposite of a fraction

1) Switch numerator and denominator 2) Change sign

(1a9) Reciprocal of a fraction

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

(1a10) Divide Fractions

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

(1b7) The square root of a fraction

1) The answer is in the form of a fraction

(1b6) What is a square root?

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

(1a7) Multiply negatives

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

(3-1) Solve linear equations

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.

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

2(x+24)

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

2(x+24)

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

216 with an exponent of 1/3

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

2x + 48

(2-3) Combine like terms

2y + 2y = 4y

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

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)

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

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)

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

5(2x-1)

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

5x + 45 - 3x + 3

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

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)

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

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

(2-24) x * 8

8x

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

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

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

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

(3) Rational Expression

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.)

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

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

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

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.

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

Distribute a negative 1

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

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

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

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

(2-10) Divide variables with exponents

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

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

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

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

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.

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

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

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

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

(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)

They are in order.

Add or subtract variables with exponents

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

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

Write the variable or base without an exponent

(2-25) Coefficient

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