Exercise 123 Page 423

a To simplify the given expression, we will use the Properties of Exponents. For this exercise, we will begin by rewriting powers with fractional exponents as roots of their bases. Let's do it!

(9^(1/2)x^2y)(27^(1/3)y^(- 1))

a^(1/2)=sqrt(a)

(sqrt(9)x^2y)(27^(1/3)y^(- 1))

PowToRootSL

a^(1/n)=sqrt(a)

(sqrt(9)x^2y)(sqrt(27)y^(- 1))

CalcRoot

Calculate root

(3x^2y)(3y^(- 1))

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Simplify

SplitIntoFactors

Split into factors

3* x^2 * y * 3 * y^(- 1)

CommutativePropMult

Commutative Property of Multiplication

3 * 3 * x^2 * y^(- 1) * y

Multiply

9* x^2 * y^(- 1) * y

NegOneExponentToFrac

a^(- 1)=1/a

9* x^2 * 1/y * y

FracMultDenomToNumber

a/y* y = a

9* x^2 * 1

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Simplify

IdPropMult

Identity Property of Multiplication

9* x^2

Multiply

9x^2

b To simplify the given expression we will use the Properties of Exponents. We can start by applying the Power of a Power Property.

( x^(1/2) )^(- 2)

PowPow

(a^m)^n=a^(m* n)

x^(12(- 2))

MultPosNeg

a(- b)=- a * b

x^(- 12(2))

FracMultDenomToNumber

a/2* 2 = a

x^(- 1)

Now, notice that this expression has a negative exponent. When this is the case, the variable can be moved to the denominator and the exponent will become positive. a^(- n)= 1/a^n ⇒ x^(- 1)=1/x

c To simplify the given expression, remember that the numerator of a rational exponent is the exponent of the expression, and the denominator is the index.

a^()1 n=sqrt(a) and a^() m n=sqrt(a^m) Let's simplify the expression!

( 1/125)^(23)

MoveNumRight

a/b=1/b* a

( 1/125)^(13 * 2)

ProdInExponent

a^(m* n)=(a^m)^n

(( 1/125)^(13))^2

PowToRootD

a^(1n)=sqrt(a)

(sqrt(1/125))^2

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