To simplify the expression [tex]\(\left(3 a^{-4} b^5\right)^{-4}\)[/tex] and ensure that all exponents are positive, we will use exponent rules in a step-by-step process.
1. Apply the exponent rule for products:
The expression inside the parentheses can be decomposed as [tex]\((3 a^{-4} b^5)^{-4}\)[/tex].
2. Apply the power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
Each component inside the parentheses must be raised to the power of [tex]\(-4\)[/tex].
[tex]\[
(3 a^{-4} b^5)^{-4} = 3^{-4} \cdot (a^{-4})^{-4} \cdot (b^5)^{-4}
\][/tex]
3. Simplify each term separately:
- For the constant [tex]\(3\)[/tex]:
[tex]\[
3^{-4}
\][/tex]
- For [tex]\(a^{-4}\)[/tex]:
[tex]\[
(a^{-4})^{-4} = a^{(-4) \cdot (-4)} = a^{16}
\][/tex]
- For [tex]\(b^5\)[/tex]:
[tex]\[
(b^5)^{-4} = b^{5 \cdot (-4)} = b^{-20}
\][/tex]
4. Combine all the simplified terms:
[tex]\[
3^{-4} \cdot a^{16} \cdot b^{-20}
\][/tex]
5. Convert any negative exponents to positive exponents:
- [tex]\(3^{-4}\)[/tex] can be written as [tex]\(\frac{1}{3^4}\)[/tex].
- [tex]\(b^{-20}\)[/tex] can be written as [tex]\(\frac{1}{b^{20}}\)[/tex].
6. Simplify further if needed:
[tex]\[
3^{-4} = \frac{1}{3^4} = \frac{1}{81}
\][/tex]
So the expression [tex]\(\left(3 a^{-4} b^5\right)^{-4} \)[/tex] simplifies to:
[tex]\[
\frac{a^{16}}{81 b^{20}}
\][/tex]
Hence, the simplified expression using positive exponents is:
[tex]\[
\frac{a^{16}}{81 b^{20}}
\][/tex]
