Use the rules of exponents to simplify the following expression. Assume that all variables are nonzero.

Question

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:

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