To simplify the expression [tex]\(\left(27 a^6 b^{-9}\right)^{2 / 3}\)[/tex] and express the answer with positive exponents only, we can follow these steps:
1. Rewrite the expression inside the parentheses:
[tex]\[
(27 \cdot a^6 \cdot b^{-9})^{2/3}
\][/tex]
2. Apply the exponent [tex]\(\frac{2}{3}\)[/tex] to each term inside the parentheses:
[tex]\[
27^{2/3} \cdot (a^6)^{2/3} \cdot (b^{-9})^{2/3}
\][/tex]
3. Simplify each term separately:
- For [tex]\(27^{2/3}\)[/tex]:
[tex]\[
27^{2/3} = (3^3)^{2/3} = 3^{3 \cdot \frac{2}{3}} = 3^2 = 9
\][/tex]
- For [tex]\((a^6)^{2/3}\)[/tex]:
[tex]\[
(a^6)^{2/3} = a^{6 \cdot \frac{2}{3}} = a^4
\][/tex]
- For [tex]\((b^{-9})^{2/3}\)[/tex]:
[tex]\[
(b^{-9})^{2/3} = b^{-9 \cdot \frac{2}{3}} = b^{-6}
\][/tex]
4. Combine the simplified terms:
[tex]\[
9 \cdot a^4 \cdot b^{-6}
\][/tex]
5. Convert [tex]\(b^{-6}\)[/tex] to a positive exponent:
[tex]\[
b^{-6} = \frac{1}{b^6}
\][/tex]
6. Combine everything into a single fraction:
[tex]\[
9 \cdot a^4 \cdot \frac{1}{b^6} = \frac{9a^4}{b^6}
\][/tex]
Therefore, the simplified expression with positive exponents only is:
[tex]\[
\left(27 a^6 b^{-9}\right)^{2 / 3} = \frac{9a^4}{b^6}
\][/tex]
