To solve the expression [tex]\(\frac{5 x^2}{\sqrt{5 x^2}}\)[/tex], we will simplify it step by step.
1. Identify the Numerator and the Denominator:
The numerator of the given expression is [tex]\(5 x^2\)[/tex].
The denominator of the given expression is [tex]\(\sqrt{5 x^2}\)[/tex].
2. Simplify the Denominator:
Let's simplify [tex]\(\sqrt{5 x^2}\)[/tex]:
[tex]\[
\sqrt{5 x^2} = \sqrt{5} \cdot \sqrt{x^2}
\][/tex]
Since [tex]\(\sqrt{x^2} = |x|\)[/tex] (the absolute value of [tex]\(x\)[/tex]), we have:
[tex]\[
\sqrt{5 x^2} = \sqrt{5} \cdot |x|
\][/tex]
3. Rewrite the Expression:
Substituting the simplified denominator back into the original expression, we get:
[tex]\[
\frac{5 x^2}{\sqrt{5} \cdot |x|}
\][/tex]
4. Simplify the Fraction:
Notice that [tex]\(5 x^2\)[/tex] in the numerator can be written as [tex]\(5 \cdot x \cdot x\)[/tex]:
[tex]\[
\frac{5 \cdot x \cdot x}{\sqrt{5} \cdot |x|}
\][/tex]
We can separate the fraction as follows:
[tex]\[
\frac{5}{\sqrt{5}} \cdot \frac{x \cdot x}{|x|}
\][/tex]
Simplify each part separately:
- Simplifying [tex]\(\frac{5}{\sqrt{5}}\)[/tex]:
[tex]\[
\frac{5}{\sqrt{5}} = \frac{5 \cdot \sqrt{5}}{5} = \sqrt{5}
\][/tex]
- Simplifying [tex]\(\frac{x \cdot x}{|x|} = \frac{x^2}{|x|}\)[/tex]:
Note that [tex]\(\frac{x^2}{|x|}\)[/tex] simplifies to [tex]\(|x|\)[/tex], since:
[tex]\[
\frac{x^2}{|x|} = \frac{x \cdot x}{|x|} = \frac{x \cdot |x|}{|x|} = |x|
\][/tex]
Putting it all together:
[tex]\[
\frac{5 x^2}{\sqrt{5 x^2}} = \sqrt{5} \cdot |x|
\][/tex]
5. Final Answer:
The simplified expression is:
[tex]\[
\boxed{\sqrt{5} \cdot |x|}
\][/tex]
