Question
Use the rules of exponents to simplify the following expression. Assume that all variables are nonzero.
left parenthesis StartFraction x Superscript negative 5 Baseline y Over z EndFraction right parenthesis Superscript negative 6
Question content area bottom
Part 1
left parenthesis StartFraction x Superscript negative 5 Baseline y Over z EndFraction right parenthesis Superscript negative 6
equals
enter your response here
(Simplify your answer. Use integers or fractions for any numbers in the expression. Use positive exponents only.)
left parenthesis StartFraction x Superscript negative 5 Baseline y Over z EndFraction right parenthesis Superscript negative 6
Question content area bottom
Part 1
left parenthesis StartFraction x Superscript negative 5 Baseline y Over z EndFraction right parenthesis Superscript negative 6
equals
enter your response here
(Simplify your answer. Use integers or fractions for any numbers in the expression. Use positive exponents only.)
Answers
GPT-4o mini
To simplify the expression
\[
\left( \frac{x^{-5} y}{z} \right)^{-6},
\]
we will use the rules of exponents. Specifically, we will use the property that \((a/b)^{-n} = \frac{b^n}{a^n}\) and the property that \((a^m)^n = a^{m \cdot n}\).
First, we apply the negative exponent rule:
\[
\left( \frac{x^{-5} y}{z} \right)^{-6} = \frac{z^6}{(x^{-5} y)^6}.
\]
Next, we'll simplify the denominator \((x^{-5} y)^6\):
\[
(x^{-5} y)^6 = (x^{-5})^6 \cdot (y^6) = x^{-30} y^6.
\]
Now we can rewrite the entire expression:
\[
\frac{z^6}{x^{-30} y^6}.
\]
To handle the negative exponent in the denominator, we can rewrite it as:
\[
\frac{z^6}{x^{-30} y^6} = z^6 \cdot \frac{1}{x^{-30} y^6} = z^6 \cdot \frac{1}{y^6} \cdot x^{30}.
\]
This can be expressed as:
\[
z^6 \cdot x^{30} \cdot \frac{1}{y^6} = \frac{x^{30} z^6}{y^6}.
\]
Thus, the simplified expression is:
\[
\frac{x^{30} z^6}{y^6}.
\]
Therefore, the final answer is:
\[
\frac{x^{30} z^6}{y^6}.
\]
\[
\left( \frac{x^{-5} y}{z} \right)^{-6},
\]
we will use the rules of exponents. Specifically, we will use the property that \((a/b)^{-n} = \frac{b^n}{a^n}\) and the property that \((a^m)^n = a^{m \cdot n}\).
First, we apply the negative exponent rule:
\[
\left( \frac{x^{-5} y}{z} \right)^{-6} = \frac{z^6}{(x^{-5} y)^6}.
\]
Next, we'll simplify the denominator \((x^{-5} y)^6\):
\[
(x^{-5} y)^6 = (x^{-5})^6 \cdot (y^6) = x^{-30} y^6.
\]
Now we can rewrite the entire expression:
\[
\frac{z^6}{x^{-30} y^6}.
\]
To handle the negative exponent in the denominator, we can rewrite it as:
\[
\frac{z^6}{x^{-30} y^6} = z^6 \cdot \frac{1}{x^{-30} y^6} = z^6 \cdot \frac{1}{y^6} \cdot x^{30}.
\]
This can be expressed as:
\[
z^6 \cdot x^{30} \cdot \frac{1}{y^6} = \frac{x^{30} z^6}{y^6}.
\]
Thus, the simplified expression is:
\[
\frac{x^{30} z^6}{y^6}.
\]
Therefore, the final answer is:
\[
\frac{x^{30} z^6}{y^6}.
\]