To simplify the expression [tex]\(\left(-3 y^2 z^{-3}\right)^4\)[/tex] and write the result using only positive exponents, follow these steps:
1. Distribute the exponent to each factor inside the parentheses:
The given expression is [tex]\(\left(-3 y^2 z^{-3}\right)^4\)[/tex]. We need to raise each part of the product to the power of 4:
[tex]\[
\left(-3 y^2 z^{-3}\right)^4 = (-3)^4 (y^2)^4 (z^{-3})^4
\][/tex]
2. Calculate each part separately:
- [tex]\((-3)^4\)[/tex]:
[tex]\[
(-3)^4 = (-3) \times (-3) \times (-3) \times (-3) = 81
\][/tex]
- [tex]\((y^2)^4\)[/tex]:
[tex]\[
(y^2)^4 = y^{2 \times 4} = y^8
\][/tex]
- [tex]\((z^{-3})^4\)[/tex]:
[tex]\[
(z^{-3})^4 = z^{-3 \times 4} = z^{-12}
\][/tex]
3. Combine the results:
Now, combine the results from the separate parts:
[tex]\[
(-3)^4 (y^2)^4 (z^{-3})^4 = 81 y^8 z^{-12}
\][/tex]
4. Write the final answer using only positive exponents:
The term [tex]\(z^{-12}\)[/tex] has a negative exponent. To convert it to a positive exponent, we can write it as a reciprocal:
[tex]\[
z^{-12} = \frac{1}{z^{12}}
\][/tex]
Therefore, the final expression using only positive exponents is:
[tex]\[
81 y^8 z^{-12} = 81 y^8 \cdot \frac{1}{z^{12}} = \frac{81 y^8}{z^{12}}
\][/tex]
Thus, the simplified expression using only positive exponents is:
[tex]\[
\boxed{\frac{81 y^8}{z^{12}}}
\][/tex]
