Question

which expression is equivalent to \\(\sqrt{\frac{2x^5}{18}}\\)? assume \\(x \geq 0\\).
options:
\\(\frac{x^2\sqrt{x}}{3}\\)
\\(\frac{\sqrt{x}}{3x^2}\\)
\\(\frac{3\sqrt{x}}{x^2}\\)
\\(\frac{2x\sqrt{x}}{3}\\)

Explanation:

Step1: Simplify the square root

We start with the expression \(\sqrt{\frac{2x^5}{18}}\). First, simplify the fraction inside the square root: \(\frac{2x^5}{18}=\frac{x^5}{9}\). So now we have \(\sqrt{\frac{x^5}{9}}\).

Step2: Apply square root properties

Using the property \(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\) (for \(a\geq0, b > 0\)), we can rewrite this as \(\frac{\sqrt{x^5}}{\sqrt{9}}\). We know that \(\sqrt{9} = 3\), and for \(\sqrt{x^5}\), we can use the property \(\sqrt{x^n}=x^{\frac{n}{2}}\) (for \(x\geq0\)). So \(x^5=x^{4 + 1}=x^4\cdot x\), and \(\sqrt{x^5}=\sqrt{x^4\cdot x}=\sqrt{x^4}\cdot\sqrt{x}=x^2\sqrt{x}\) (since \(\sqrt{x^4}=x^2\) for \(x\geq0\)).

Step3: Combine the terms

Now we substitute back into our expression: \(\frac{x^2\sqrt{x}}{3}\).

Answer:

\(\frac{x^2\sqrt{x}}{3}\) (which is the top - middle option, the second option in the vertical list of options)