Chapter 4.2, Problem 14OE

To determine

If the statement $a^3\cdot a^2=a^6$ is true then give the law of exponents that justify the given statement else give a counter example.

Answer to Problem 14OE

The statement $a^3\cdot a^2=a^6$ is false.

Explanation of Solution

Given information: The given statement is,

  $a^3\cdot a^2=a^6\mathrm{......}\left(1\right)$

Calculation: The given statement $a^3\cdot a^2=a^6$ is false because of the formula ${\left(x\right)}^m\cdot {\left(x\right)}^n={\left(x\right)}^{m+n}$ . Therefore the correct statement will be $a^3\cdot a^2=a^5$ .

For example consider a counter example for which $a=2$ .

Now satisfy $a=2$ in (1),

  $\begin{array}{c}{\left(2\right)}^3\cdot {\left(2\right)}^2={\left(2\right)}^6\mathrm{......}\left(1\right)\mathrm{....}\left(\text{F}ora=2\right)\\ \left(8\right)\cdot \left(4\right)=64\\ 32=\mathrm{64....}\left(\text{Which is false}\right)\end{array}$