Determine whether the statement is true or false. $$ \sqrt{x^{2}}=|x| $$

Question

UpStudy AI · Accepted Answer

1. We start with the definition of the square root function. For any nonnegative real number $$ a $$, the square root $$ \sqrt{a} $$ is defined as the unique nonnegative number $$ b $$ such that $$ b^2 = a $$.

2. Consider any real number $$ x $$. Squaring $$ x $$ gives $$ x^2 $$, which is always nonnegative. Thus, $$ x^2 \geq 0 $$.

3. Applying the square root to $$ x^2 $$ gives:
   $$
   \sqrt{x^2}
   $$
   By the definition of the square root, $$ \sqrt{x^2} $$ is the unique nonnegative number whose square is $$ x^2 $$.

4. Notice that the absolute value $$ |x| $$ is defined as:
   $$
   |x| =
   \begin{cases}
   x, & \text{if } x \geq 0, \
   -x, & \text{if } x < 0.
   \end{cases}
   $$
   In both cases, $$ |x| $$ is always nonnegative, and it satisfies $$ |x|^2 = x^2 $$.

5. Since both $$ \sqrt{x^2} $$ and $$ |x| $$ are the unique nonnegative numbers whose square is $$ x^2 $$, we conclude:
   $$
   \sqrt{x^2} = |x|
   $$

The statement is true.
The statement $$ \sqrt{x^{2}} = |x| $$ is true.

Determine whether the statement is true or false.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Algebra Questions