Sure, let's solve the expression step-by-step.
The objective is to simplify the expression inside the square root: [tex]\(\sqrt{25 x^2 - 7 x^2}\)[/tex].
Step 1: Simplify the expression inside the square root.
[tex]\[
25 x^2 - 7 x^2
\][/tex]
Combine the like terms:
[tex]\[
(25 - 7)x^2 = 18 x^2
\][/tex]
Step 2: Substitute the simplified expression back into the square root.
[tex]\[
\sqrt{18 x^2}
\][/tex]
Step 3: Simplify the square root expression.
Recall that [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex].
In this case, we can break it down as:
[tex]\[
\sqrt{18 x^2} = \sqrt{18} \cdot \sqrt{x^2}
\][/tex]
Step 4: Simplify [tex]\(\sqrt{x^2}\)[/tex] and [tex]\(\sqrt{18}\)[/tex].
[tex]\(\sqrt{x^2} = |x|\)[/tex] (the absolute value of [tex]\(x\)[/tex]), and [tex]\(\sqrt{18}\)[/tex] can be simplified further as:
[tex]\[
\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3 \sqrt{2}
\][/tex]
Therefore, combining these results, we get:
[tex]\[
\sqrt{18 x^2} = 3 \sqrt{2} \cdot |x|
\][/tex]
Thus, the simplified form of [tex]\(\sqrt{25 x^2 - 7 x^2}\)[/tex] is [tex]\(3 \sqrt{2} \cdot |x|\)[/tex].
In this context, since we are given a specific outcome, it's important to consider that the presented result:
[tex]\[
\sqrt{25 x^2 - 7 x^2} = 3 \sqrt{2} \cdot \sqrt{x^2}
\][/tex]
is equivalent to:
[tex]\[
3 \sqrt{2} \cdot |x|
\][/tex]
So, the final answer is:
[tex]\[
\boxed{3 \sqrt{2} \cdot |x|}
\][/tex]
