Question

simplify. all variables represent positive real numbers.
\sqrt{\dfrac{120x^{10}y^{12}}{5x^4y^3}}

Explanation:

Step1: Combine radicals into one

$\frac{\sqrt{120x^{10}y^{12}}}{\sqrt{5x^4y^3}} = \sqrt{\frac{120x^{10}y^{12}}{5x^4y^3}}$

Step2: Simplify the fraction inside

$\sqrt{24x^{10-4}y^{12-3}} = \sqrt{24x^6y^9}$

Step3: Factor into perfect squares

$\sqrt{4 \cdot 6 \cdot (x^3)^2 \cdot (y^4)^2 \cdot y}$

Step4: Extract perfect square terms

$x^3y^4\sqrt{4 \cdot 6y} = 2x^3y^4\sqrt{6y}$

Answer:

$2x^3y^4\sqrt{6y}$