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:

$$\boxed{{\displaystyle x^{12}{y}^3{z}^3}}$$