Final answer:
To simplify √ ( (128x⁵y⁶)/(2x⁷y⁵) ), divide numerically and algebraically to get √(64y/x²), which simplifies to 8y/x.
Step-by-step explanation:
To find the simplified form of √ ( (128x⁵y⁶)/(2x⁷y⁵) ), we first need to simplify the expression under the square root by dividing the numerators and the denominators. We start by simplifying the numeric part 128/2 which equals to 64. Next, we simplify the variables by subtracting the exponents of like bases since we are dividing. For x⁵/x⁷, we get x⁵-⁷ which simplifies to x⁻² or 1/x². For y⁶/y⁵, the simplification is y⁶-⁵ which equals to y.
Now that we have the simplified form of the expression under the square root, we have √(64y/x²). Taking the square root of 64 gives us 8, and the square root of y would just be y since it is already a perfect square. But, for x², the square root is x since x² equals x multiplied by itself.
The final simplified form after taking the square root is therefore 8y/x.