Question

Simplify${\left(\frac{8a^3{b}^2}{16a^2{b}^3}\right)}^4$ assuming that no variable equal to 0 .

Short answer

The simplified value of expression is $\frac{a^4{b}^{-4}}{16}$.

Step-by-step solution

Step 1. Write down the given expression.

The given expression to simplify is ${\left(\frac{8a^3{b}^2}{16a^2{b}^3}\right)}^4$.

Step 2. Simplify the given expression.

Simplifying ${\left(\frac{8a^3{b}^2}{16a^2{b}^3}\right)}^4$gives,

$\begin{array}{c}{\left(\frac{8a^3{b}^2}{16a^2{b}^3}\right)}^4={\left(\frac{a^3{b}^2}{2a^2{b}^3}\right)}^4\\ =\frac{{\left(a^3{b}^2\right)}^4}{{\left(2\right)}^4{\left(a^2{b}^3\right)}^4}\\ =\frac{\left(a^{12}{b}^8\right)}{16\left(a^8{b}^{12}\right)}\\ =\frac{\left(a^{12-8}{b}^{8-12}\right)}{16}\mathrm{....}(\text{Using}\frac{{\left(x\right)}^a{\left(x\right)}^b}{{\left(x\right)}^c}=x^{a+b-c})\\ =\frac{a^4{b}^{-4}}{16}\end{array}$

Step 3. Conclusion.

The simplified value of expression is $\frac{a^4{b}^{-4}}{16}$.