Certainly! Let's simplify the expression [tex]\(\left(5 x^{-4} p^4\right)^{-2}\)[/tex] and write it using positive exponents.
1. Initial Expression:
[tex]\[
\left(5 x^{-4} p^4\right)^{-2}
\][/tex]
2. Distribute the Exponent [tex]\(-2\)[/tex]:
We apply the exponent [tex]\(-2\)[/tex] to each factor inside the parentheses:
[tex]\[
\left(5 x^{-4} p^4\right)^{-2} = 5^{-2} \cdot (x^{-4})^{-2} \cdot (p^4)^{-2}
\][/tex]
3. Simplify Each Term:
- For [tex]\(5^{-2}\)[/tex]:
[tex]\[
5^{-2} = \frac{1}{5^2} = \frac{1}{25}
\][/tex]
- For [tex]\((x^{-4})^{-2}\)[/tex]:
[tex]\[
(x^{-4})^{-2} = x^{(-4) \cdot (-2)} = x^8
\][/tex]
- For [tex]\((p^4)^{-2}\)[/tex]:
[tex]\[
(p^4)^{-2} = p^{4 \cdot (-2)} = p^{-8} = \frac{1}{p^8}
\][/tex]
4. Combine the Simplified Terms:
Now, multiply the simplified terms together:
[tex]\[
5^{-2} \cdot (x^{-4})^{-2} \cdot (p^4)^{-2} = \frac{1}{25} \cdot x^8 \cdot \frac{1}{p^8}
\][/tex]
Combine the fractions:
[tex]\[
= \frac{x^8}{25} \cdot \frac{1}{p^8} = \frac{x^8}{25 p^8}
\][/tex]
Thus, the simplified expression with positive exponents is:
[tex]\[
\frac{x^8}{25 p^8}
\][/tex]
