To simplify the expression [tex]\(\left(-4 x^5 y^{-2}\right)^2\)[/tex], we need to apply the rules of exponents step by step. Here is the detailed process:
1. Apply the power to each factor inside the parentheses:
The expression given is [tex]\(\left(-4 x^5 y^{-2}\right)^2\)[/tex]. According to the power of a product rule [tex]\((ab)^n = a^n \cdot b^n\)[/tex], we can distribute the exponent 2 to each component inside the parentheses:
[tex]\[
\left(-4 x^5 y^{-2}\right)^2 = (-4)^2 \cdot (x^5)^2 \cdot (y^{-2})^2
\][/tex]
2. Simplify each component separately:
- For [tex]\((-4)^2\)[/tex]:
[tex]\[
(-4)^2 = 16
\][/tex]
- For [tex]\((x^5)^2\)[/tex], apply the power of a power rule [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[
(x^5)^2 = x^{5 \cdot 2} = x^{10}
\][/tex]
- For [tex]\((y^{-2})^2\)[/tex], again apply the power of a power rule:
[tex]\[
(y^{-2})^2 = y^{-2 \cdot 2} = y^{-4}
\][/tex]
3. Combine the simplified components:
Now, we put all the simplified parts together:
[tex]\[
(-4)^2 \cdot (x^5)^2 \cdot (y^{-2})^2 = 16 \cdot x^{10} \cdot y^{-4}
\][/tex]
4. Write the final simplified expression:
The simplified form of the given expression is:
[tex]\[
16 x^{10} y^{-4}
\][/tex]
Alternatively, we can also express [tex]\(y^{-4}\)[/tex] as [tex]\(\frac{1}{y^4}\)[/tex] if preferred:
[tex]\[
16 x^{10} y^{-4} = \frac{16 x^{10}}{y^4}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
16 x^{10} y^{-4}
\][/tex]
