Chapter 4.2, Problem 18OE

To determine

If the statement ${\left(a^{2}b\right)}^2=a^2{b}^2$ is true then give the law of exponents that justify the given statement else give a counter example.

Answer to Problem 18OE

The statement ${\left(a^{2}b\right)}^2=a^2{b}^2$ is false.

Explanation of Solution

Given information: The given statement is,

  ${\left(a^{2}b\right)}^2=a^2{b}^2\mathrm{......}\left(1\right)$

Calculation: The given statement ${\left(a^{2}b\right)}^2=a^2{b}^2$ is false because of law of exponents ${\left(xy\right)}^n={\left(x\right)}^n{\left(y\right)}^n$ . Therefore from (1), ${\left(a^{2}b\right)}^2$ should be equal to $a^4{b}^2$ .

However from (1) ${\left(a^{2}b\right)}^2=a^2{b}^2$ which is false. Hence the given statement is false. For example consider a counter example for which $a=2$ and $b=3$ . Therefore from (1),

  $\begin{array}{l}{\left({\left(2\right)}^2\left(3\right)\right)}^2={\left(2\right)}^2{\left(3\right)}^2\mathrm{......}\left(a=2,b=3\right)\\ {\left(12\right)}^2=36\\ 144=\mathrm{36....}\left(\text{Which is false}\right)\end{array}$