To simplify the expression [tex]\((3 x^{-6} y^3)^{-3}\)[/tex] and express the answer using only positive exponents, follow these steps:
1. Distribute the exponent to each term inside the parentheses:
[tex]\[
(3 x^{-6} y^3)^{-3} = 3^{-3} \cdot (x^{-6})^{-3} \cdot (y^3)^{-3}
\][/tex]
2. Simplify each term separately:
- For [tex]\(3^{-3}\)[/tex]:
[tex]\[
3^{-3} = \frac{1}{3^3} = \frac{1}{27}
\][/tex]
- For [tex]\((x^{-6})^{-3}\)[/tex]:
[tex]\[
(x^{-6})^{-3} = x^{-6 \times -3} = x^{18}
\][/tex]
- For [tex]\((y^3)^{-3}\)[/tex]:
[tex]\[
(y^3)^{-3} = y^{3 \times -3} = y^{-9}
\][/tex]
3. Combine the simplified terms:
[tex]\[
\frac{1}{27} \cdot x^{18} \cdot y^{-9}
\][/tex]
4. Express the term with a negative exponent as a fraction with a positive exponent:
[tex]\[
\frac{1}{27} \cdot x^{18} \cdot \frac{1}{y^9} = \frac{x^{18}}{27y^9}
\][/tex]
So, the simplified expression using only positive exponents is:
[tex]\[
\boxed{\frac{x^{18}}{27y^9}}
\][/tex]
