Which of the following represents the fully simplified form of the algebraic expression be… — Armed Services Vocational Aptitude Battery (ASVAB)

Which of the following represents the fully simplified form of the algebraic expression below, assuming $x \neq 0$ and $y \neq 0$ ?

\left( \frac{2x^{-3}y^4}{4x^2y^{-1}} \right)^{-2}

A
$\frac{y^{10}}{4x^{10}}$
B
$\frac{4x^2}{y^6}$
$\frac{4x^{10}}{y^{10}}$ Answer
D
$4x^{12}y^8$

Answer

The expression simplifies to

\frac{4x^{10}}{y^{10}}

The correct answer is derived by first applying the quotient rule to simplify the expression inside the parentheses, being careful to subtract negative exponents correctly (e.g.,

y^{4 - (-1)} = y^5

). Then, the negative exponent on the outside of the parentheses means taking the reciprocal of the fraction and squaring all terms, which applies to the coefficient (2 becomes 4) and the variables (

x^5

becomes

x^{10}

Step-by-Step Solution

Simplify the numerical coefficients inside the parentheses.

\frac{2}{4} = \frac{1}{2}

Simplifying the fraction before handling variables reduces computational complexity.

Apply the quotient rule for exponents to the variables

x

and

y

For

x

-3 - 2 = -5

. For

y

4 - (-1) = 4 + 1 = 5

. The expression inside the parentheses becomes

\frac{1}{2} x^{-5}y^5

\frac{y^5}{2x^5}

When dividing terms with the same base, subtract the exponent of the denominator from the exponent of the numerator.

Rewrite the entire expression with the simplified inner term.

\left( \frac{y^5}{2x^5} \right)^{-2}

This establishes the simplified base expression before applying the final outer exponent.

Apply the negative exponent property to the fraction.

\left( \frac{2x^5}{y^5} \right)^2

A negative exponent on a fraction indicates the reciprocal of that fraction raised to the positive exponent.

Apply the power of a quotient and product rules.

\frac{2^2(x^5)^2}{(y^5)^2} = \frac{4x^{10}}{y^{10}}

When raising a power to a power, multiply the exponents, and ensure the outer exponent is applied to the numerical coefficient.

Key Concept

Operations with Variable Exponents

Estimated Time:2m 0s

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