For all real numbers x, the identity \sqrt{x^2} = x holds true. — GMAT

For all real numbers $x$ , the identity $\sqrt{x^2} = x$ holds true.

Answer: Answer

The statement is false because the principal square root

\sqrt{x^2}

is defined as the absolute value of

x

|x|

, which equals

-x

when

x < 0

The statement is false because the expression

\sqrt{x^2}

represents the principal (non-negative) square root of

x^2

, which simplifies to

|x|

. When

x

is negative,

|x| = -x

, which is not equal to

x

Recall the definition of the principal square root radical symbol

\sqrt{\vphantom{x}}

By definition,

\sqrt{a} \ge 0

for any real number

a \ge 0

The principal square root function always yields a non-negative output.

Test a negative value for the variable

x

Let

x = -3

. Then

\sqrt{(-3)^2} = \sqrt{9} = 3

Evaluating a specific negative number tests if the equality holds universally for all real numbers.

Compare the evaluated result with the original value of

x

Since

3 \neq -3

\sqrt{x^2} \neq x

when

x < 0

A single counterexample disproves a universal mathematical identity statement.

Principal Square Root and Absolute Value Property (\sqrt{x^2} = |x|)

Estimated Time:45s

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