To determine which expression is equivalent to [tex]\(\sqrt{\frac{2 x^5}{18}}\)[/tex], we will start by simplifying the given expression step-by-step. Here's the detailed process:
1. Original Expression:
[tex]\[
\sqrt{\frac{2 x^5}{18}}
\][/tex]
2. Simplify the Fraction Inside the Square Root:
Simplify the fraction inside the square root by dividing the numerator and the denominator by their greatest common divisor, which is 2:
[tex]\[
\frac{2 x^5}{18} = \frac{2 x^5}{2 \cdot 9} = \frac{x^5}{9}
\][/tex]
3. Rewrite the Expression with Simplified Fraction:
Substitute the simplified fraction back into the square root:
[tex]\[
\sqrt{\frac{x^5}{9}}
\][/tex]
4. Separate the Square Root:
Use the property of square roots that states [tex]\(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\)[/tex]:
[tex]\[
\sqrt{\frac{x^5}{9}} = \frac{\sqrt{x^5}}{\sqrt{9}}
\][/tex]
5. Simplify the Denominator:
The square root of 9 is 3:
[tex]\[
\sqrt{9} = 3
\][/tex]
So, the expression becomes:
[tex]\[
\frac{\sqrt{x^5}}{3}
\][/tex]
6. Simplify the Numerator:
Recall that [tex]\(\sqrt{x^5}\)[/tex] can be written as [tex]\(x^{5/2}\)[/tex]. The exponent 5/2 means [tex]\(x\)[/tex] to the power of 5 and then taking the square root:
[tex]\[
\sqrt{x^5} = x^{5/2}
\][/tex]
Therefore, the expression simplifies to:
[tex]\[
\frac{x^{5/2}}{3}
\][/tex]
7. Rewrite [tex]\(x^{5/2}\)[/tex]:
We can rewrite [tex]\(x^{5/2}\)[/tex] as [tex]\(x^2 \cdot x^{1/2}\)[/tex]:
[tex]\[
x^{5/2} = x^2 \cdot x^{1/2}
\][/tex]
8. Substitute Back in the Expression:
So, the final simplified form of our expression is:
[tex]\[
\frac{x^2 \cdot x^{1/2}}{3}
\][/tex]
or equivalently:
[tex]\[
\frac{x^2 \sqrt{x}}{3}
\][/tex]
Now, we compare this simplified form with the given options. The correct expression among the options is:
[tex]\[
\frac{x^2 \sqrt{x}}{3}
\][/tex]
So, the expression equivalent to [tex]\(\sqrt{\frac{2 x^5}{18}}\)[/tex] is:
[tex]\[
\boxed{\frac{x^2 \sqrt{x}}{3}}
\][/tex]
