Question

Select the equivalent expression.

\[\left(\dfrac{a^{-3}}{b^{2}}\right)^{4}=?\]
Choose 1 answer:
Choose 1 answer:
(Choice A)

\[\dfrac{b^2}{a^7}\]
A

\[\dfrac{b^2}{a^7}\]
(Choice B)

\[\dfrac{1}{a^{12}\cdot b^{8}}\]
B

\[\dfrac{1}{a^{12}\cdot b^{8}}\]
(Choice C)

\[\left( \dfrac{b}{a}\right)^{20}\]
C

\[\left( \dfrac{b}{a}\right)^{20}\]

Answer 1

To simplify the expression \(\left(\dfrac{a^{-3}}{b^{2}}\right)^{4}\), we can use the property of exponents \((\frac{x}{y})^n = \frac{x^n}{y^n}\).

1. Start by applying the exponent to both the numerator and the denominator:
\[
\left(\dfrac{a^{-3}}{b^{2}}\right)^{4} = \dfrac{(a^{-3})^{4}}{(b^{2})^{4}}
\]

2. Now calculate the exponents:
\[
(a^{-3})^{4} = a^{-12}
\]
\[
(b^{2})^{4} = b^{8}
\]

3. Substitute these back into the fraction:
\[
\dfrac{(a^{-3})^{4}}{(b^{2})^{4}} = \dfrac{a^{-12}}{b^{8}}
\]

4. Since \(a^{-12} = \frac{1}{a^{12}}\), we can rewrite this as:
\[
\dfrac{a^{-12}}{b^{8}} = \dfrac{1}{a^{12} \cdot b^{8}}
\]

So the equivalent expression is:
\[
\dfrac{1}{a^{12} \cdot b^{8}}
\]

The correct choice is:
(Choice B) \(\dfrac{1}{a^{12}\cdot b^{8}}\)