Questions: True or false, if x is any real number, then sqrt(x^2) = x

Transcript text: True or false, if $x$ is any real number, then $\sqrt{x^{2}}=x$

Solution

Solution Steps

Step 1: Understanding the Statement

The statement claims that for any real number $ x $, $ \sqrt{x^{2}} = x $. We need to determine if this is always true.

Step 2: Analyzing the Square Root Function

The square root function $ \sqrt{y} $ is defined to return the non-negative value of $ y $. Therefore, $ \sqrt{x^{2}} $ will always return the non-negative value of $ x^{2} $.

Step 3: Testing with a Negative Number

Let $ x = -3 $. Then: \[ \sqrt{(-3)^{2}} = \sqrt{9} = 3 \] Here, $ \sqrt{x^{2}} = 3 $, but $ x = -3 $. Thus, $ \sqrt{x^{2}} \neq x $ when $ x $ is negative.

Step 4: Conclusion

Since the statement does not hold true for negative values of $ x $, the statement is false.