Option D:
[tex]\frac{25 a^{9} b^{10} }{6 }[/tex] is equivalent to the given expression.
Solution:
Given expression:
[tex]$\frac{(5 a b)^{3}}{30 a^{-6} b^{-7}}[/tex]
To find which expression is equivalent to the given expression.
[tex]$\frac{(5 a b)^{3}}{30 a^{-6} b^{-7}}[/tex]
Using exponent rule: [tex](ab)^m=a^mb^m[/tex]
[tex]$=\frac{5^3 a^3 b^{3}}{30 a^{-6} b^{-7}}[/tex]
Using exponent rule: [tex]\frac{1}{a^{m}}=a^{-m}, \quad \frac{1}{a^{-m}}=a^{m}[/tex]
[tex]$=\frac{125 a^3 b^{3}a^{6} b^{7}}{30 }[/tex]
[tex]$=\frac{125 a^3 a^{6} b^{3} b^{7}}{30 }[/tex]
Using exponent rule: [tex]a^{m} \cdot a^{n}=a^{m+n}[/tex]
[tex]$=\frac{125 a^{3+6} b^{3+7} }{30 }[/tex]
[tex]$=\frac{125 a^{9} b^{10} }{30 }[/tex]
Divide both numerator and denominator by the common factor 5.
[tex]$=\frac{25 a^{9} b^{10} }{6 }[/tex]
[tex]$\frac{(5 a b)^{3}}{30 a^{-6} b^{-7}}=\frac{25 a^{9} b^{10} }{6 }[/tex]
Therefore, [tex]\frac{25 a^{9} b^{10} }{6 }[/tex] is equivalent to the given expression.
Hence Option D is the correct answer.
