Sure! Let's solve the given expression step-by-step:
We are given the expression [tex]\(\sqrt{5x \cdot \left[(x+5)^2 - (x-5)^2\right]}\)[/tex].
1. First, let's simplify the expression inside the square root, which is [tex]\(5x \cdot \left[(x+5)^2 - (x-5)^2\right]\)[/tex].
2. Consider the part inside the parentheses: [tex]\((x+5)^2 - (x-5)^2\)[/tex]. This is a difference of squares and can be expanded:
[tex]\[
(x+5)^2 = x^2 + 10x + 25
\][/tex]
[tex]\[
(x-5)^2 = x^2 - 10x + 25
\][/tex]
Subtracting these two expressions, we get:
[tex]\[
(x^2 + 10x + 25) - (x^2 - 10x + 25) = x^2 + 10x + 25 - x^2 + 10x - 25
\][/tex]
Simplifying further:
[tex]\[
x^2 - x^2 + 10x + 10x + 25 - 25 = 20x
\][/tex]
Therefore:
[tex]\[
(x+5)^2 - (x-5)^2 = 20x
\][/tex]
3. Substitute this back into the original expression:
[tex]\[
5x \cdot 20x = 100x^2
\][/tex]
4. Now our expression inside the square root simplifies to [tex]\(100x^2\)[/tex]:
[tex]\[
\sqrt{100x^2}
\][/tex]
5. Simplify the square root of [tex]\(100x^2\)[/tex]:
[tex]\[
\sqrt{100x^2} = 10\sqrt{x^2}
\][/tex]
6. Since [tex]\(\sqrt{x^2} = |x|\)[/tex], we finally have:
[tex]\[
10|x|
\][/tex]
Given the context and assuming [tex]\(x \geq 0\)[/tex] for simplicity, we can write [tex]\(10|x| = 10x\)[/tex].
So, [tex]\(\sqrt{5x \cdot \left[(x+5)^2 - (x-5)^2\right]} = 10x\)[/tex].
In summary, the step-by-step solution yields:
[tex]\[
\sqrt{5x \cdot \left[(x+5)^2 - (x-5)^2\right]} = 100x^2, \quad \sqrt{100x^2} = 10x \quad \text{if} \ x \geq 0.
\][/tex]
