If the expression \frac{(p^3 q^{-2})^{-1}}{(p^{-1} q^2)^3} is simplified to the form p^x q… — ACT

If the expression $\frac{(p^3 q^{-2})^{-1}}{(p^{-1} q^2)^3}$ is simplified to the form $p^x q^y$ for all non-zero real numbers $p$ and $q$ , what is the value of $2x - y$ ?

Answer: 4

Answer

The correct answer is 4.

Applying the power of a power rule to the numerator yields

(p^3 q^{-2})^{-1} = p^{-3} q^2

. Applying the same rule to the denominator yields

(p^{-1} q^2)^3 = p^{-3} q^6

. Dividing the terms by subtracting exponents gives

p^{-3 - (-3)} q^{2-6} = p^0 q^{-4}

. Thus,

x = 0

and

y = -4

. Evaluating

2x - y

gives

2(0) - (-4) = 4

Step-by-Step Solution

Simplify the numerator using the power of a power property, which states that

(a^m)^n = a^{mn}

(p^3 q^{-2})^{-1} = p^{3 \cdot (-1)} q^{-2 \cdot (-1)} = p^{-3} q^2

This distributes the exponent of

-1

to both factors inside the parentheses in the numerator.

Simplify the denominator using the power of a power property, which states that

(a^m)^n = a^{mn}

(p^{-1} q^2)^3 = p^{-1 \cdot 3} q^{2 \cdot 3} = p^{-3} q^6

This distributes the exponent of

3

to both factors inside the parentheses in the denominator.

Combine the simplified numerator and denominator using the quotient property of exponents, which states that

\frac{a^m}{a^n} = a^{m-n}

\frac{p^{-3} q^2}{p^{-3} q^6} = p^{-3 - (-3)} q^{2 - 6} = p^0 q^{-4}

This simplifies division by subtracting the exponent in the denominator from the exponent in the numerator for each base.

Identify the values of

x

and

y

from the simplified form

p^0 q^{-4}

, and evaluate the final expression

2x - y

x = 0

and

y = -4

, so

2(0) - (-4) = 4

This substitutes the values of the exponents into the target algebraic expression to find the final numerical answer.

Key Concept

Properties of exponents including power of a power and quotient properties

Alternative Method

Alternatively, you can write the terms with positive exponents first:

\frac{(p^3 q^{-2})^{-1}}{(p^{-1} q^2)^3} = \frac{(p^{-1} q^2)^3}{(p^3 q^{-2})^1} = \frac{p^{-3} q^6}{p^3 q^{-2}} = p^{-3-3} q^{6-(-2)} = p^{-6} q^8

, but this expression is equivalent to the original expression only if inverted properly. Direct distribution is less prone to inversion errors.

Estimated Time:1m 30s

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