Question

Use the rules of exponents to simplify the following expression. Assume that all variables are nonzero.
left parenthesis 4 x Superscript negative 2 Baseline y Superscript 5 Baseline right parenthesis Superscript negative 3

Answer 1

To simplify the expression \( \left( 4 x^{-2} y^{5} \right)^{-3} \), we can use the rules of exponents.

1. **Apply the power of a product rule**: \( (ab)^n = a^n b^n \)

Therefore, we can distribute the exponent \(-3\) to each factor:
\[
(4)^{-3} (x^{-2})^{-3} (y^{5})^{-3}
\]

2. **Simplify each term:**
- For \( (4)^{-3} \):
\[
(4)^{-3} = \frac{1}{4^3} = \frac{1}{64}
\]
- For \( (x^{-2})^{-3} \):
\[
(x^{-2})^{-3} = x^{-2 \cdot -3} = x^{6}
\]
- For \( (y^{5})^{-3} \):
\[
(y^{5})^{-3} = y^{5 \cdot -3} = y^{-15}
\]

3. **Combine the results**:
Putting it all together, we have:
\[
\left( 4 x^{-2} y^{5} \right)^{-3} = \frac{1}{64} x^{6} y^{-15}
\]

4. **Express the negative exponent in the denominator**:
Rewrite \( y^{-15} \) as \( \frac{1}{y^{15}} \):
\[
= \frac{x^{6}}{64 y^{15}}
\]

Thus, the simplified expression is:
\[
\frac{x^{6}}{64 y^{15}}
\]