Question

rationalize the denominator of \\(\sqrt4{dfrac{256}{9x^3}}\\). assume that all variables represent positive real numbers.

Explanation:

Step1: Rewrite the radical as a fraction of radicals

We know that \(\sqrt[4]{\frac{a}{b}}=\frac{\sqrt[4]{a}}{\sqrt[4]{b}}\), so \(\sqrt[4]{\frac{256}{9x^{3}}}=\frac{\sqrt[4]{256}}{\sqrt[4]{9x^{3}}}\)
Since \(256 = 4^4\), then \(\sqrt[4]{256}=4\), so we have \(\frac{4}{\sqrt[4]{9x^{3}}}\)

Step2: Determine the missing factors to rationalize the denominator

For a fourth - root, we want the radicand in the denominator to be a perfect fourth - power. The radicand in the denominator is \(9x^{3}=3^{2}x^{3}\). We need to multiply the numerator and denominator by \(\sqrt[4]{3^{2}x}\) (because \((3^{2}x^{3})\times(3^{2}x)=3^{4}x^{4}\), which is a perfect fourth - power).

Step3: Multiply the numerator and denominator

Multiply the numerator and denominator by \(\sqrt[4]{9x}\) (note that \(3^{2}=9\)):
\[

$$\begin{align*} \frac{4}{\sqrt[4]{9x^{3}}}\times\frac{\sqrt[4]{9x}}{\sqrt[4]{9x}}&=\frac{4\sqrt[4]{9x}}{\sqrt[4]{9x^{3}\times9x}}\\ &=\frac{4\sqrt[4]{9x}}{\sqrt[4]{81x^{4}}} \end{align*}$$

\]

Step4: Simplify the denominator

Since \(\sqrt[4]{81x^{4}}=\sqrt[4]{3^{4}x^{4}} = 3x\) (because \(x>0\))

Step5: Write the final expression

The expression becomes \(\frac{4\sqrt[4]{9x}}{3x}\)

Answer:

\(\frac{4\sqrt[4]{9x}}{3x}\)