Chapter 5.2, Problem 14OE

To determine

To find: Simplify the given expression.

Answer to Problem 14OE

  $\frac{a}{2^{-1}\cdot b^{-2}}=2a{b}^2$

Explanation of Solution

Given information:

  $\frac{a}{2^{-1}\cdot b^{-2}}$

Concept:

  $\frac{a}{b}\cdot \frac{c}{d}=\frac{a\cdot c}{b\cdot d}$

Using law of exponents

  $\begin{array}{l}{a}^{-m}=\frac{1}{a^m}\\ a^m\cdot a^n=a^{m+n}\\ \frac{a^m}{a^n}=a^{m-n}\\ a^0=1\end{array}$

  ${\left(a^n\right)}^m=a^{mn}$

Where $a$ be any value.

  $\frac{a}{2^{-1}\cdot b^{-2}}$

  $\begin{array}{l}\Rightarrow \frac{a}{2^{-1}\cdot b^{-2}}=a\cdot \frac{1}{2^{-1}\cdot b^{-2}}\\ \Rightarrow \frac{a}{2^{-1}\cdot b^{-2}}=a\cdot \frac{1}{2^{-1}}\cdot \frac{1}{b^{-2}}\\ \Rightarrow \frac{a}{2^{-1}\cdot b^{-2}}=a\cdot \frac{1}{\left(\frac{1}{2}\right)}\cdot \frac{1}{\left(\frac{1}{b^2}\right)}\\ \Rightarrow \frac{a}{2^{-1}\cdot b^{-2}}=a\cdot 2\cdot b^2\\ \Rightarrow \frac{a}{2^{-1}\cdot b^{-2}}=2a{b}^2\end{array}$

Therefore,

  $\frac{a}{2^{-1}\cdot b^{-2}}=2a{b}^2$