which expression is equivalent to $\frac{b^{-2}}{ab^{-3}}$? assume $a\neq0,b\neq0$.\n$\frac{a}{b^{5}}$\n$\fra…

which expression is equivalent to $\frac{b^{-2}}{ab^{-3}}$? assume $a\neq0,b\neq0$.\n$\frac{a}{b^{5}}$\n$\frac{1}{ab^{5}}$\n$\frac{a^{3}b}{1}$\n$\frac{b}{a}$

which expression is equivalent to $\frac{b^{-2}}{ab^{-3}}$? assume $a\neq0,b\neq0$.\n$\frac{a}{b^{5}}$\n$\frac{1}{ab^{5}}$\n$\frac{a^{3}b}{1}$\n$\frac{b}{a}$

Answer

Explanation:

Step1: Use negative - exponent rule

Recall that $x^{-n}=\frac{1}{x^{n}}$. So, $\frac{b^{-2}}{ab^{-3}}=\frac{\frac{1}{b^{2}}}{a\times\frac{1}{b^{3}}}$.

Step2: Rewrite division as multiplication

Dividing by a fraction is the same as multiplying by its reciprocal. So, $\frac{\frac{1}{b^{2}}}{a\times\frac{1}{b^{3}}}=\frac{1}{b^{2}}\times\frac{b^{3}}{a}$.

Step3: Multiply the fractions

When multiplying fractions $\frac{m}{n}\times\frac{p}{q}=\frac{mp}{nq}$. Here, $\frac{1}{b^{2}}\times\frac{b^{3}}{a}=\frac{b^{3}}{ab^{2}}$.

Step4: Use the quotient - rule of exponents

The quotient - rule states that $\frac{x^{m}}{x^{n}}=x^{m - n}$. So, $\frac{b^{3}}{ab^{2}}=\frac{b^{3-2}}{a}=\frac{b}{a}$.

Answer:

$\frac{b}{a}$