which expression is equivalent to $\frac{(5ab)^{3}}{30a^{-6}b^{-7}}$? assume $a\neq0,b\neq0$.\n$\frac{a^{7}b^…

which expression is equivalent to $\frac{(5ab)^{3}}{30a^{-6}b^{-7}}$? assume $a\neq0,b\neq0$.\n$\frac{a^{7}b^{10}}{6}$\n$\frac{125a^{18}b^{21}}{30}$\n$\frac{25a^{3}b^{4}}{6}$\n$\frac{25a^{9}b^{10}}{6}$

which expression is equivalent to $\frac{(5ab)^{3}}{30a^{-6}b^{-7}}$? assume $a\neq0,b\neq0$.\n$\frac{a^{7}b^{10}}{6}$\n$\frac{125a^{18}b^{21}}{30}$\n$\frac{25a^{3}b^{4}}{6}$\n$\frac{25a^{9}b^{10}}{6}$

Answer

Explanation:

Step1: Expand the numerator

First, expand $(5ab)^3$ using the power - of - a - product rule $(xy)^n=x^n y^n$. So, $(5ab)^3 = 5^3\times a^3\times b^3=125a^3b^3$. The original expression becomes $\frac{125a^3b^3}{30a^{- 6}b^{-7}}$.

Step2: Simplify the coefficient and use the quotient rule of exponents

Simplify the coefficient $\frac{125}{30}=\frac{25}{6}$. For the variables with the same base, use the quotient rule $\frac{x^m}{x^n}=x^{m - n}$. For $a$: $\frac{a^3}{a^{-6}}=a^{3-(-6)}=a^{3 + 6}=a^9$. For $b$: $\frac{b^3}{b^{-7}}=b^{3-(-7)}=b^{3 + 7}=b^{10}$. The simplified expression is $\frac{25a^9b^{10}}{6}$.

Answer:

$\frac{25a^9b^{10}}{6}$