To simplify the given expression [tex]\(\frac{(5ab)^3}{30a^{-6}b^{-7}}\)[/tex], let's go step-by-step:
1. Simplify the Numerator:
[tex]\[
(5ab)^3 = 5^3 \cdot a^3 \cdot b^3 = 125a^3b^3
\][/tex]
So, the new expression is:
[tex]\[
\frac{125a^3b^3}{30a^{-6}b^{-7}}
\][/tex]
2. Simplify the Denominator:
[tex]\[
30a^{-6}b^{-7}
\][/tex]
3. Combine the Constants:
[tex]\[
\frac{125}{30} = \frac{25}{6} \quad \text{(since } 125 \div 5 = 25 \text{ and } 30 \div 5 = 6\text{)}
\][/tex]
4. Simplify the Exponents for [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- For [tex]\(a\)[/tex]:
[tex]\[
a^3 \div a^{-6} = a^{3 - (-6)} = a^{3 + 6} = a^9
\][/tex]
- For [tex]\(b\)[/tex]:
[tex]\[
b^3 \div b^{-7} = b^{3 - (-7)} = b^{3 + 7} = b^{10}
\][/tex]
5. Combine Everything:
So, the entire simplified expression is:
[tex]\[
\frac{25a^9b^{10}}{6}
\][/tex]
Therefore, the expression equivalent to [tex]\(\frac{(5ab)^3}{30a^{-6}b^{-7}}\)[/tex] is:
[tex]\[
\boxed{\frac{25a^9b^{10}}{6}}
\][/tex]
