Dividing and multiplying surds - Rule

Question

What rule/process allows me to take this equation: $$6x^{2} \cdot \sqrt{\frac{y}{x}}$$

And simplify it to become: $$6x \cdot \sqrt{xy}$$

Answer 1

This simplification uses the fact that multiplication is preserved "under the radical", so to speak. That is:

$6x^{2} * \sqrt{\frac{y}{x}} = 6x * \sqrt{x^{2}} * \sqrt{\frac{y}{x}} = 6x * \sqrt{x^{2}*\frac{y}{x}} = 6x*\sqrt{xy}$

Answer 2

$6x^2 * \sqrt{y/x}= 6x *x * \sqrt{y/x}=6x * \sqrt{x}*\sqrt{x}*\sqrt{y/x}= 6x * \sqrt{xy} $

Answer 3

$\dfrac{y}{x}= \dfrac{xy}{x*x} $,$\sqrt{\dfrac{xy}{x*x}}=\dfrac{\sqrt{xy}}{\sqrt{x^2}},\sqrt{x^2}=x,\dfrac{6x^2}{x}=6x$ so you got the final answer.

Answer 4

$$\text{If} x,y> 0,\text{then},6x^{2}\sqrt{\frac{y}{x}}=6x^{2}\sqrt{\frac{xy}{x^2}}=6x^{2}\frac{\sqrt{xy}}{\sqrt x^2}=6x^{2}\frac{\sqrt{xy}}{x}=6x\sqrt{xy}$$