Question

given all variables below are positive, express in simplest radical form with a rational denominator.\\(\frac{\sqrt{336x^{10}y^{6}}}{\sqrt{7y^{3}}}\\)

Explanation:

Step1: Combine square roots

$\frac{\sqrt{336x^{10}y^6}}{\sqrt{7y^3}} = \sqrt{\frac{336x^{10}y^6}{7y^3}}$

Step2: Simplify the fraction inside

$\sqrt{\frac{336}{7} \cdot x^{10} \cdot \frac{y^6}{y^3}} = \sqrt{48x^{10}y^3}$

Step3: Factor into perfect squares

$\sqrt{16 \cdot 3 \cdot x^{10} \cdot y^2 \cdot y} = \sqrt{16x^{10}y^2} \cdot \sqrt{3y}$

Step4: Simplify perfect square roots

$4x^5y\sqrt{3y}$

Answer:

$4x^5y\sqrt{3y}$