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