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.

 

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