Question

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

Explanation:

Step1: Simplify the fraction inside the square root

First, simplify \(\frac{2x^5}{18}\) by dividing numerator and denominator by 2. We get \(\frac{x^5}{9}\). So the expression becomes \(\sqrt{\frac{x^5}{9}}\).

Step2: Use the property of square roots \(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\)

Applying this property, we have \(\frac{\sqrt{x^5}}{\sqrt{9}}\). Since \(\sqrt{9} = 3\), this simplifies to \(\frac{\sqrt{x^5}}{3}\).

Step3: Simplify \(\sqrt{x^5}\)

We can rewrite \(x^5\) as \(x^4 \cdot x\). Then \(\sqrt{x^4 \cdot x}=\sqrt{x^4} \cdot \sqrt{x}\). Since \(x\geq0\), \(\sqrt{x^4}=x^2\). So \(\sqrt{x^5}=x^2\sqrt{x}\).

Step4: Substitute back into the expression

Substituting \(\sqrt{x^5}=x^2\sqrt{x}\) into \(\frac{\sqrt{x^5}}{3}\), we get \(\frac{x^2\sqrt{x}}{3}\).

Answer:

\(\frac{x^2\sqrt{x}}{3}\) (the first option)