Question

which expression shows the simplified form of \\( \left(8r^{-5}\
ight)^{-3} \\)?

\\( 8r^{15} \\)

\\( \frac{8}{r^{15}} \\)

\\( 512r^{15} \\)

\\( \frac{r^{15}}{512} \\)

Explanation:

Distribute the outer exponent

To simplify the expression \((8r^{-5})^{-3}\), we apply the power of a product rule, which states that \((ab)^n = a^n b^n\).

$$ (8r^{-5})^{-3} = 8^{-3} \cdot (r^{-5})^{-3} $$

Apply the power of a power rule

Next, we use the power of a power rule, \((x^a)^b = x^{a \cdot b}\), to simplify the term with the variable \(r\).

$$ (r^{-5})^{-3} = r^{-5 \cdot (-3)} = r^{15} $$

Simplify the numerical coefficient

Now we evaluate the numerical coefficient \(8^{-3}\) using the negative exponent rule, \(x^{-n} = \frac{1}{x^n}\).

$$ 8^{-3} = \frac{1}{8^3} = \frac{1}{512} $$

Combine the simplified terms

We multiply the simplified coefficient and the variable term together to get the final simplified expression.

$$ \frac{1}{512} \cdot r^{15} = \frac{r^{15}}{512} $$

Answer:

• (A) \(8r^{15}\)
• (B) \(\frac{8}{r^{15}}\)
• (C) \(512r^{15}\)
• (D) \(\frac{r^{15}}{512}\) (Correct answer)