Question

question
express in simplest radical form assuming $x > 0$.
\\(\sqrt{3x^{11}}\\)

answer
\\(\circ\\) \\(\sqrt{3x^{11}}\\) \\(\circ\\) \\(3x^5\\)
\\(\circ\\) \\(x^5\sqrt{3x}\\) \\(\circ\\) \\(x^6\sqrt{3x}\\)

Explanation:

Step1: Split exponent of $x$

Rewrite $x^{11}$ as $x^{10} \cdot x$, since $10$ is the largest even number less than $11$.
$$\sqrt{3x^{11}} = \sqrt{3 \cdot x^{10} \cdot x}$$

Step2: Separate perfect square factor

Use property $\sqrt{ab}=\sqrt{a}\sqrt{b}$ to isolate the perfect square $x^{10}$.
$$\sqrt{3 \cdot x^{10} \cdot x} = \sqrt{x^{10}} \cdot \sqrt{3x}$$

Step3: Simplify perfect square root

Since $\sqrt{x^{10}} = x^5$ (as $x>0$, $(x^5)^2=x^{10}$).
$$\sqrt{x^{10}} \cdot \sqrt{3x} = x^5\sqrt{3x}$$

Answer:

$\boldsymbol{x^5\sqrt{3x}}$ (corresponding to the option: $x^5\sqrt{3x}$)