If a and b are non-zero real numbers, which of the following expressions is equivalent to … — ACT

If $a$ and $b$ are non-zero real numbers, which of the following expressions is equivalent to $\frac{a^4 b^{-2}}{(a^2 b^{-3})^2}$ ?

b^4

To simplify the expression, first apply the power of a product rule to the denominator:

(a^2 b^{-3})^2 = (a^2)^2 (b^{-3})^2 = a^4 b^{-6}

. Next, divide the numerator by the simplified denominator by subtracting the exponents of the corresponding bases:

\frac{a^4 b^{-2}}{a^4 b^{-6}} = a^{4-4} b^{-2 - (-6)} = a^0 b^4

. Since

a^0 = 1

for any non-zero real number

a

, the expression simplifies to

b^4

Simplify the denominator using the power of a product and power of a power rules.

(a^2 b^{-3})^2 = (a^2)^2 \cdot (b^{-3})^2 = a^4 b^{-6}

When raising a product to a power, raise each factor to that power. When raising a power to a power, multiply the exponents.

Substitute the simplified denominator back into the original fraction.

\frac{a^4 b^{-2}}{a^4 b^{-6}}

To prepare the expression for division.

Divide the numerator by the denominator by subtracting the exponents of like bases.

a^{4-4} b^{-2 - (-6)} = a^0 b^4 = 1 \cdot b^4 = b^4

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

Properties of exponents, specifically the power of a product rule, power of a power rule, and quotient rule.

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