Select the equivalent expression.

Question

\[\left(x^{-3}\cdot y^{3}\right)^{-7}=?\]
Choose 1 answer:
Choose 1 answer:
(Choice A)
\[x^{-10}\cdot y^{-4}\]
A
\[x^{-10}\cdot y^{-4}\]
(Choice B)
\[(xy)^0\]
B
\[(xy)^0\]
(Choice C)
\[x^{21}\cdot y^{-21}\]
C
\[x^{21}\cdot y^{-21}\]

Answer 1

To simplify the expression \(\left(x^{-3}\cdot y^{3}\right)^{-7}\), we will apply the properties of exponents.

1. Start by applying the power of a product rule, which states that \((ab)^n = a^n \cdot b^n\).

\[ \left(x^{-3} \cdot y^{3}\right)^{-7} = \left(x^{-3}\right)^{-7} \cdot \left(y^{3}\right)^{-7} \]

2. Now, we can simplify each component. Using the power of a power rule, which states that \((a^m)^n = a^{m \cdot n}\):

\[ \left(x^{-3}\right)^{-7} = x^{-3 \cdot (-7)} = x^{21} \]

\[ \left(y^{3}\right)^{-7} = y^{3 \cdot (-7)} = y^{-21} \]

3. Combining these results, we have:

\[ \left(x^{-3} \cdot y^{3}\right)^{-7} = x^{21} \cdot y^{-21} \]

So the equivalent expression is:

\[ x^{21} \cdot y^{-21} \]

Thus, the answer is:

Choice C: \[x^{21}\cdot y^{-21}\]