Questions: Which expression is equivalent to sqrt(128 x^5 y^6 / 2 x^7 y^5) ? Assume x>0 and y>0. x sqrt(y) / 8 y sqrt(x) / 8 8 sqrt(x) / y 8 sqrt(y) / x

Solution Steps

Start by simplifying the fraction inside the square root: $$\sqrt{\frac{128 x^{5} y^{6}}{2 x^{7} y^{5}}}$$ Divide the coefficients and subtract the exponents of like bases: $$\frac{128}{2} = 64, \quad x^{5-7} = x^{-2}, \quad y^{6-5} = y^{1}$$ So the expression becomes: $$\sqrt{64 x^{-2} y}$$

Break down the square root into separate parts: $$\sqrt{64 x^{-2} y} = \sqrt{64} \cdot \sqrt{x^{-2}} \cdot \sqrt{y}$$ Simplify each part: $$\sqrt{64} = 8, \quad \sqrt{x^{-2}} = x^{-1} = \frac{1}{x}, \quad \sqrt{y} = \sqrt{y}$$ Combine the simplified parts: $$8 \cdot \frac{1}{x} \cdot \sqrt{y} = \frac{8 \sqrt{y}}{x}$$

The simplified expression is: $$\frac{8 \sqrt{y}}{x}$$ Compare this with the provided options:

1. $\frac{x \sqrt{y}}{8}$
2. $\frac{y \sqrt{x}}{8}$
3. $\frac{8 \sqrt{x}}{y}$
4. $\frac{8 \sqrt{y}}{x}$

The correct match is option 4.

Final Answer