Question

Use a combination of rules for exponents to simplify the expression to exponential form with only positive exponents. Assume variables represent nonzero real numbers.
left parenthesis StartFraction 2 x Superscript negative 3 Baseline y Over 3 Superscript negative 1 Baseline z squared EndFraction right parenthesis Superscript negative 3
Question content area bottom
Part 1
What does left parenthesis StartFraction 2 x Superscript negative 3 Baseline y Over 3 Superscript negative 1 Baseline z squared EndFraction right parenthesis Superscript negative 3
​equal?
A.StartFraction 2 y cubed Over 3 x Superscript 9 Baseline z Superscript 6 EndFraction
StartFraction 2 y cubed Over 3 x Superscript 9 Baseline z Superscript 6 EndFraction
B.StartFraction x Superscript 9 Baseline z Superscript 6 Over 216 y cubed EndFraction
StartFraction x Superscript 9 Baseline z Superscript 6 Over 216 y cubed EndFraction
C.StartFraction 27 x Superscript 9 Baseline y cubed Over 8 z Superscript 6 EndFraction
StartFraction 27 x Superscript 9 Baseline y cubed Over 8 z Superscript 6 EndFraction
D.StartFraction 3 xyz Superscript 12 Over 2 z EndFraction
StartFraction 3 xyz Superscript 12 Over 2 z EndFraction

Answer 1

To simplify the expression \( \left( \frac{2x^{-3}y}{3^{-1}z^2} \right)^{-3} \), we will follow the rules of exponents.

1. **Reciprocal of Negative Exponents:** Recall that \( a^{-n} = \frac{1}{a^n} \). Therefore, we can rewrite the expression with positive exponents inside by moving \( x^{-3} \) and \( 3^{-1} \) to the denominator and numerator, respectively.

2. Rewrite the base:
\[
\frac{2x^{-3}y}{3^{-1}z^2} = \frac{2y}{3^{-1}z^2} \cdot \frac{1}{x^3} = \frac{2y \cdot 3}{z^2} \cdot \frac{1}{x^3} = \frac{6y}{x^3z^2}
\]

Thus,
\[
\left( \frac{2x^{-3}y}{3^{-1}z^2} \right)^{-3} = \left( \frac{6y}{x^3 z^2} \right)^{-3}
\]

3. **Apply the negative exponent:** From the rule \( \left(\frac{a}{b}\right)^{-n} = \frac{b^n}{a^n} \), we apply the exponent \( -3 \):
\[
= \frac{(x^3z^2)^3}{(6y)^3}
\]
which expands to:
\[
= \frac{x^{9}z^{6}}{216y^{3}}
\]
(since \( 6^3 = 216 \)).

4. **Final result with positive exponents:** Therefore, the expression simplifies to:
\[
\frac{x^{9}z^{6}}{216y^{3}}
\]

Now, matching it to the options provided:

B. \( \frac{x^{9}z^{6}}{216y^{3}} \)

Thus, the correct answer is:

**B. \( \frac{x^{9}z^{6}}{216y^{3}} \)**