(z+2)^9(4z-1)^7/(z+2)^(10)(4z-1)^5 To simplify this expression we can use that a power divided by another power with the same base can be simplified by subtracting the exponents.

(z+2)^9(4z-1)^7/(z+2)^(10)(4z-1)^5

WriteProdFrac

Write as a product of fractions

(z+2)^9/(z+2)^(10)*(4z-1)^7/(4z-1)^5

DivPow

a^m/a^n= a^(m-n)

(z+2)^(-1)(4z-1)^2

NegOneExponentToFrac

a^(- 1)=1/a

1/(z+2)*(4z-1)^2

MoveRightFacToNumOne

1/b* a = a/b

(4z-1)^2/z+2

d The numerator and denominator of this equation are not factored completely.

(x+2)(x^2-6x+9)/(x-3)(x^2-4) Let's factor the all of the expressions in the numerator and denominator before simplifying.

x^2-6x+9

FacNegPerfectSquare

a^2-2ab+b^2=(a-b)^2

(x-3)^2

PowToProdTwoFac

a^2=a* a

(x-3)(x-3)

Now let's factor the expression from the denominator.

x^2-4

WritePow

Write as a power

x^2-2^2

FacDiffSquares

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

(x+2)(x-2)

The second factor in the denominator can be factored to (x+2)(x-2). We can use these factored forms to simplify the original expression.

(x+2)(x^2-6x+9)/(x-3)(x^2-4)

SubstituteExpressions

Substitute expressions

(x+2)(x-3)(x-3)/(x-3)(x+2)(x-2)

CommutativePropMult

Commutative Property of Multiplication

(x+2)(x-3)(x-3)/(x+2)(x-3)(x-2)

WriteProdFrac

Write as a product of fractions

(x+2)(x-3)/(x+2)(x-3)*x-3/x-2

QuotOne

a/a=1

1*x-3/x-2

IdPropMult

Identity Property of Multiplication

x-3/x-2

Subchapter links

3.2.1 Investigating Rational Functions p.140-141

3.2.2 Simplifying Rational Expressions p.145-146

3.2.3 Multiplying and Dividing Rational Expressions p.148-149

3.2.4 Adding and Subtracting Rational Expressions p.153-154

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3.2.5 Creating New Functions p.157-159

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Exercise 105 Page 153

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