Finding root a^2

Finding $\sqrt{ a^{2}}$

In the expression $\sqrt{ a^{2}}$ , the radicand is a perfect square. It is tempting to think

that $\sqrt{ a^{2}}$ = a, but we see below that this is not the case.

Suppose a = 5. Then we have $\sqrt{ 5^{2}}$ , which is $\sqrt{25}$ , or 5.

Suppose a = -5. Then we have $\sqrt{ -5^{2}}$ ,which is $\sqrt{25}$ , or 5.

Suppose a = 0 . Then we have $\sqrt{ 0^{2}}$ which is $\sqrt{0}$ , or 0.

The symbol $\sqrt{ a^{2}}$ never represents a negative number. It represents the

principal square root of a². Note the following.

SIMPLIFYING $\sqrt{ a^{2}}$

a ≥ 0 $\rightarrow \sqrt{ a^{2}}$ = a

if a is positive or 0 ,the principal square root of $\sqrt(a)$ is a.

a < 0 $\rightarrow \sqrt{ a^{2}}$ = -a

If a is negative, the principal square root of $\sqrt(a)$ is the opposite of a.

In all cases, the radical expression represents the absolute value of a.

PRINCIPAL SQUARE ROOT OF a²

For any real number a, $\sqrt{ a^{2}}$ = \left | a \right | . The principal (nonnegative) square

root of a² is the absolute value of a.