Question

given all variables below are positive, express in simplest radical form with a rational denominator.
$\frac{sqrt{135x^{8}y^{13}}}{sqrt{5y^{4}}}$

Explanation:

Step1: Combine square roots

$\frac{\sqrt{135x^8y^{13}}}{\sqrt{5y^4}} = \sqrt{\frac{135x^8y^{13}}{5y^4}}$

Step2: Simplify fraction inside root

$\sqrt{\frac{135}{5} \cdot x^8 \cdot \frac{y^{13}}{y^4}} = \sqrt{27x^8y^9}$

Step3: Factor into perfect squares

$\sqrt{9 \cdot 3 \cdot x^8 \cdot y^8 \cdot y} = \sqrt{9x^8y^8 \cdot 3y}$

Step4: Extract perfect squares

$\sqrt{9x^8y^8} \cdot \sqrt{3y} = 3x^4y^4\sqrt{3y}$

Answer:

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