Question

use the quotient rule to simplify. assume that all variables represent positive real numbers.
\\(\sqrt3{\frac{4}{x^6}}\\)
\\(\sqrt3{\frac{4}{x^6}} = \square\\)
(type an exact answer, using radicals as needed. simplify your answer.)

Explanation:

Step1: Recall the quotient rule for radicals

The quotient rule for cube roots states that $\sqrt[3]{\frac{a}{b}}=\frac{\sqrt[3]{a}}{\sqrt[3]{b}}$ for any real numbers $a$ and $b$ (where $b
eq0$). So we can apply this rule to $\sqrt[3]{\frac{4}{x^{6}}}$:
$\sqrt[3]{\frac{4}{x^{6}}}=\frac{\sqrt[3]{4}}{\sqrt[3]{x^{6}}}$

Step2: Simplify the denominator

We know that for a cube root, $\sqrt[3]{x^{n}} = x^{\frac{n}{3}}$. For $n = 6$, we have $\sqrt[3]{x^{6}}=x^{\frac{6}{3}}=x^{2}$.

Step3: Write the final simplified form

Substituting the simplified denominator back into the fraction, we get $\frac{\sqrt[3]{4}}{x^{2}}$.

Answer:

$\frac{\sqrt[3]{4}}{x^{2}}$