Question

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

Answer 1

To simplify the expression \(\left(\frac{x^{-4} y}{z}\right)^{-3}\), we will apply the rules of exponents.

1. **Apply the negative exponent rule**: \((a/b)^{-n} = (b/a)^{n}\).

So, we start by rewriting the expression:

\[
\left(\frac{x^{-4} y}{z}\right)^{-3} = \frac{z^3}{(x^{-4} y)^3}
\]

2. **Distribute the exponent to the numerator and denominator**:

\[
= \frac{z^3}{(x^{-4})^3 \cdot y^3}
\]

3. **Simplify the powers**: Use the power of a power rule \((a^m)^n = a^{m \cdot n}\):

\[
(x^{-4})^3 = x^{-12}
\]

So now, we have:

\[
= \frac{z^3}{x^{-12} \cdot y^3}
\]

4. **Rewrite the expression using the property of negative exponents**: Recall that \(a^{-n} = \frac{1}{a^n}\):

\[
= \frac{z^3}{\frac{1}{x^{12} y^3}} = z^3 \cdot \frac{x^{12} y^3}{1} = z^3 x^{12} y^3
\]

5. **Combine the expression**:

\[
= x^{12} y^3 z^3
\]

Thus, the simplified expression is:

\[
\boxed{x^{12} y^3 z^3}
\]