Chapter A.10, Problem 117AYU

To determine

To find: the example to show that $\sqrt{a^2}$ is not equal to $a$ and explanation for $\sqrt{a^2}=\left|a\right|$ .

Answer to Problem 117AYU

The example and the explanation is shown below.

Explanation of Solution

Given information:

The given term is $\sqrt{a^2}$ and $\sqrt{a^2}=\left|a\right|$ .

Calculation:

Consider a term $\sqrt{{\left(6\right)}^2}$

The solution will be $\sqrt{{\left(6\right)}^2}=6$

But we know that $\sqrt{{\left(-6\right)}^2}=6$

So here by above example it is clear that $\sqrt{a^2}$ is not equal to $a$ .

The square root of $a$ is defined only when $a$ is greater than or equal to zero.

Non negative result is only considered for the absolute value that’s why modulus is used.

So $\sqrt{a^2}=\left|a\right|$ .

Therefore, it is clear that $\sqrt{a^2}$ is not equal to $a$ and the reason for $\sqrt{a^2}=\left|a\right|$ is that only the absolute value is considered.