Exercise 44 Page 532

We want to simplify the given rational expression. To do so, let's recall the Negative Exponent Property. x^(- n)=1/x^n This property can be rewritten for cases where we have a negative exponent in the denominator. 1/x^(- n)=x^nLet's use the above to start simplifying our expression.

54x^3y^(-1)/3x^(-2)y

CommutativePropMult

Commutative Property of Multiplication

54x^3y^(-1)/3yx^(-2)

WriteProdFrac

Write as a product of fractions

54x^3y^(-1)/3y*1/x^(-2)

MovePartNumRight

a* b/c=a/c* b

54x^3/3y* y^(-1)*1/x^(-2)

Using the property for both factors, we can rewrite the rearranged expression. 54x^3/3y* y^(-1)*1/x^(-2) ⇕ 54x^3/3y* 1/y^1 * x^2 Now, we will split the numerator of the expression into factors. Then, we will cancel out any common factors.

54x^3/3y* 1/y^1 * x^2

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Simplify

ExponentOne

a^1=a

54x^3/3y* 1/y * x^2

MultFrac

Multiply fractions

54x^3/3y* y * x^2

MoveRightFacToNum

a/c* b = a* b/c

54x^3* x^2/3y* y

SplitIntoFactors

Split into factors

3* 18x^3* x^2/3y* y

CancelCommonFac

Cancel out common factors

3* 18x^3* x^2/3y* y

SimpQuot

Simplify quotient

18x^3* x^2/y* y

MultPow

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

18x^5/y* y

ProdToPowTwoFac

a* a=a^2

18x^5/y^2

We simplified the given expression. Finally, we will identify the restrictions on the variables from the denominator of the simplified expression and from any other denominator used.

Denominator	Restrictions on the Denominator	Restrictions on the Variables
3x^(-2)y	x^2≠ 0 and y≠ 0	x≠ 0 and y≠ 0
y^2	y^2≠ 0	y≠ 0

We found two unique restrictions on the variables. x≠ 0, y≠ 0