Question

what is the simplified form of the following expression? assume x > 0
\\(\sqrt4{\frac{3}{2x}}\\)
\\(\circ\\) \\(\frac{\sqrt4{6x}}{2x}\\)
\\(\circ\\) \\(\frac{\sqrt4{24x^3}}{2x}\\)
\\(\circ\\) \\(\frac{\sqrt4{24x^3}}{16x^4}\\)
\\(\circ\\) \\(\sqrt4{12x^2}\\)

Explanation:

Step1: Rationalize the Denominator

To simplify \(\sqrt[4]{\frac{3}{2x}}\), we need to make the denominator a perfect fourth power. Multiply the numerator and denominator inside the fourth root by \(2^3x^3\) (since we need the denominator \(2x\) to become \((2x)^4 = 16x^4\), so we multiply by \(2^3x^3\) to get the fourth power).

$$ \sqrt[4]{\frac{3}{2x}}=\sqrt[4]{\frac{3\times2^3x^3}{2x\times2^3x^3}} $$

Step2: Simplify the Fraction Inside the Root

Simplify the denominator: \(2x\times2^3x^3 = 2^{1 + 3}x^{1+3}=2^4x^4=(2x)^4\). The numerator: \(3\times2^3x^3 = 3\times8x^3 = 24x^3\). So now we have:

$$ \sqrt[4]{\frac{24x^3}{(2x)^4}} $$

Step3: Apply the Fourth Root Property

Using the property \(\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}\) (for \(b>0\)), we get:

$$ \frac{\sqrt[4]{24x^3}}{\sqrt[4]{(2x)^4}} $$

Since \(x>0\), \(\sqrt[4]{(2x)^4}=2x\). So the simplified form is \(\frac{\sqrt[4]{24x^3}}{2x}\).

Answer:

\(\frac{\sqrt[4]{24x^3}}{2x}\) (the second option)