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

Question

left parenthesis StartFraction x Superscript negative 5 Baseline y Over z EndFraction right parenthesis Superscript negative 2
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Part 1
left parenthesis StartFraction x Superscript negative 5 Baseline y Over z EndFraction right parenthesis Superscript negative 2
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​(Simplify your answer. Use integers or fractions for any numbers in the expression. Use positive exponents​ only.)

Answer 1

To simplify the expression \(\left(\frac{x^{-5}y}{z}\right)^{-2}\), we can apply the rules of exponents step by step.

1. Apply the power of a quotient rule \((a/b)^{-m} = \frac{b^m}{a^m}\): \[ \left(\frac{x^{-5}y}{z}\right)^{-2} = \frac{z^{2}}{(x^{-5}y)^{2}} \]

2. Now simplify the numerator: \[ z^{2} \text{ remains unchanged.} \]

3. Now simplify the denominator \((x^{-5}y)^{2}\) using the power of a product rule \((ab)^m = a^m b^m\): \[ (x^{-5}y)^{2} = (x^{-5})^{2}(y)^{2} = x^{-10}y^{2} \]

4. Now substitute back into the expression: \[ \frac{z^{2}}{x^{-10}y^{2}} \]

5. Now apply the rule for negative exponents \(a^{-m} = \frac{1}{a^{m}}\): \[ \frac{z^{2}}{x^{-10}y^{2}} = \frac{z^{2}y^{2}}{x^{-10}} = z^{2}y^{2}x^{10} \]

6. Finally, we will write the expression with positive exponents: \[ z^{2}y^{2}x^{10} \]

So the simplified expression is: \[ x^{10}y^{2}z^{2} \]