To solve the given expression [tex]\(\frac{(5 a b)^3}{30 a^{-6} b^{-7}}\)[/tex], we can break it down into smaller steps:
1. Calculate the numerator: [tex]\((5ab)^3\)[/tex]
[tex]\[
(5ab)^3 = 5^3 \cdot a^3 \cdot b^3 = 125a^3b^3
\][/tex]
2. Simplify the denominator: [tex]\(30a^{-6}b^{-7}\)[/tex]
[tex]\[
30a^{-6}b^{-7} = 30 \cdot \frac{1}{a^6} \cdot \frac{1}{b^7} = \frac{30}{a^6 b^7}
\][/tex]
3. Combine the numerator and the denominator:
[tex]\[
\frac{(5 a b)^3}{30 a^{-6} b^{-7}} = \frac{125a^3b^3}{\frac{30}{a^6 b^7}} = 125a^3b^3 \cdot \frac{a^6b^7}{30} = \frac{125a^3b^3 \cdot a^6b^7}{30}
\][/tex]
4. Simplify the expression:
[tex]\[
\frac{125a^3b^3 \cdot a^6b^7}{30} = \frac{125 \cdot a^{3+6} \cdot b^{3+7}}{30} = \frac{125a^9b^{10}}{30}
\][/tex]
5. Reduce the fraction if possible:
[tex]\[
\frac{125a^9b^{10}}{30}
\][/tex]
Since [tex]\(125\)[/tex] and [tex]\(30\)[/tex] have a common factor of [tex]\(5\)[/tex], we can simplify the fraction:
[tex]\[
\frac{125}{30} = \frac{25}{6}
\][/tex]
So, putting it all together:
[tex]\[
\frac{125a^9b^{10}}{30} = \frac{25a^9b^{10}}{6}
\][/tex]
Therefore, the equivalent expression is:
[tex]\[
\frac{25 a^9 b^{10}}{6}
\][/tex]
Among the given choices, the correct option is:
[tex]\(\boxed{\frac{25 a^9 b^{10}}{6}}\)[/tex]
The answer is:
[tex]\(\frac{25 a^9 b^{10}}{6}\)[/tex] which corresponds to option [tex]\(4\)[/tex].
