Question

Please help me!!! 30 minutes left

Answer 1

Option D:

$\frac{25 a^{9} b^{10} }{6 }$ is equivalent to the given expression.

Solution:

Given expression:

$\frac{(5 a b)^{3}}{30 a^{-6} b^{-7}}$

To find which expression is equivalent to the given expression.

$\frac{(5 a b)^{3}}{30 a^{-6} b^{-7}}$

Using exponent rule: $(ab)^m=a^mb^m$

    $=\frac{5^3 a^3 b^{3}}{30 a^{-6} b^{-7}}$

Using exponent rule: $\frac{1}{a^{m}}=a^{-m}, \quad \frac{1}{a^{-m}}=a^{m}$

     $=\frac{125 a^3 b^{3}a^{6} b^{7}}{30 }$

     $=\frac{125 a^3 a^{6} b^{3} b^{7}}{30 }$

Using exponent rule: $a^{m} \cdot a^{n}=a^{m+n}$

     $=\frac{125 a^{3+6} b^{3+7} }{30 }$

    $=\frac{125 a^{9} b^{10} }{30 }$

Divide both numerator and denominator by the common factor 5.

    $=\frac{25 a^{9} b^{10} }{6 }$

$\frac{(5 a b)^{3}}{30 a^{-6} b^{-7}}=\frac{25 a^{9} b^{10} }{6 }$

Therefore,  $\frac{25 a^{9} b^{10} }{6 }$ is equivalent to the given expression.

Hence Option D is the correct answer.