Sure, let's simplify the expression [tex]\(\left(-2 w^{-2} x^3\right)^5\)[/tex].
We start by distributing the exponent to each factor inside the parentheses. The expression inside the parentheses is:
[tex]\[
-2 w^{-2} x^3
\][/tex]
We need to raise each part to the 5th power:
[tex]\[
\left(-2\right)^5 \left(w^{-2}\right)^5 \left(x^3\right)^5
\][/tex]
Let's evaluate each part separately.
1. For [tex]\(\left(-2\right)^5\)[/tex]:
[tex]\[
\left(-2\right)^5 = -32
\][/tex]
2. For [tex]\(\left(w^{-2}\right)^5\)[/tex]:
[tex]\[
\left(w^{-2}\right)^5 = w^{-2 \cdot 5} = w^{-10}
\][/tex]
3. For [tex]\(\left(x^3\right)^5\)[/tex]:
[tex]\[
\left(x^3\right)^5 = x^{3 \cdot 5} = x^{15}
\][/tex]
Now we combine these results:
[tex]\[
\left(-2 w^{-2} x^3\right)^5 = -32 \cdot w^{-10} \cdot x^{15}
\][/tex]
To express the result with only positive exponents, we rewrite [tex]\(w^{-10}\)[/tex] as [tex]\(\frac{1}{w^{10}}\)[/tex]:
[tex]\[
-32 \cdot w^{-10} \cdot x^{15} = -32 \cdot \frac{x^{15}}{w^{10}}
\][/tex]
So the final simplified expression is:
[tex]\[
\frac{-32 x^{15}}{w^{10}}
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\boxed{\frac{-32 x^{15}}{w^{10}}}
\][/tex]
