To simplify the expression [tex]\(\sqrt{100 x^3}\)[/tex], let's break it down step-by-step.
The expression inside the square root is [tex]\(100 x^3\)[/tex].
1. Start by using the property of square roots that states [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]. This allows us to split the square root into two parts:
[tex]\[
\sqrt{100 x^3} = \sqrt{100} \cdot \sqrt{x^3}
\][/tex]
2. Next, simplify each square root separately:
- [tex]\(\sqrt{100}\)[/tex] is a perfect square. It simplifies to [tex]\(10\)[/tex], because [tex]\(10^2 = 100\)[/tex].
- [tex]\(\sqrt{x^3}\)[/tex] can be simplified using the property of exponents under a square root, [tex]\(\sqrt{x^n} = x^{n/2}\)[/tex]:
[tex]\[
\sqrt{x^3} = x^{3/2}
\][/tex]
3. Combine these simplified parts:
[tex]\[
\sqrt{100 x^3} = 10 \cdot x^{3/2}
\][/tex]
4. So, the simplified expression is:
[tex]\[
10 x^{3/2}
\][/tex]
However, none of the provided answer choices completely match this simplified expression. Therefore, we might reinterpret the multiple choices provided:
- [tex]\(10x\)[/tex]
- [tex]\(20x\)[/tex]
Considering [tex]\( \sqrt{100 x^3} \)[/tex] simplifies as [tex]\(10 x^{3/2}\)[/tex], and comparing the available options, the closest simplified expression that matches none directly but represents the closest comprehension in the form of using '10' factor multiplying:
Thus the correct representation is not listed in given options properly but considering multiple choice isn't complete reflection definitely or else but only [tex]\(10\sqrt{x^3}\)[/tex] seems close expectancy:
In comprehensive simplified form, should be noted: [tex]\(10\sqrt{x^3} \approx\)[/tex]
Thus the closest option might be but not exactly right among given answers but indicated expression should be [tex]\(10\sqrt{x^3}\)[/tex] on likewise expected:
So correct answer is distinctly actual value derived is :
[tex]\( 10\sqrt{x^3} \)[/tex]
