Problem:
Find the values of x and y given the equations:
√x * y = 16
x * √y = 25

Solution:

To find the values of x and y, we can solve the given equations simultaneously. Let's break down the solution into steps:

Step 1: Simplify the equations:
We can simplify the given equations by removing the square roots:

√x * y = 16 --> y = 16 / √x (Equation 1)
x * √y = 25 --> x = 25 / √y (Equation 2)

Step 2: Substitute the simplified equations into each other:
Now, we substitute the simplified forms of x and y into the respective equations:

y = 16 / √x
x = 25 / √y

Substituting y in Equation 2:

x = 25 / √(16 / √x)

Step 3: Simplify the equation:
To simplify the equation, we need to rationalize the denominator:

x = 25 / (√16 / √x)
x = 25 * (√x / √16)
x = 25 * (√x / 4)

Step 4: Solve for x:
To solve for x, we can cross-multiply:

x * 4 = 25 * √x
4x = 25√x

Step 5: Square both sides of the equation:
To eliminate the square root, we square both sides of the equation:

(4x)^2 = (25√x)^2
16x^2 = 625x

Step 6: Simplify the equation:
We simplify the equation by moving all the terms to one side:

16x^2 - 625x = 0

Step 7: Factorize and solve for x:
We can factorize the equation:

x(16x - 625) = 0

Setting each factor equal to zero:

x = 0 or 16x - 625 = 0

If x = 0, it will make y equal to 0 as well, which contradicts the given equations.

Therefore, we solve for the second factor:

16x - 625 = 0
16x = 625
x = 625 / 16
x = 39.0625

Step 8: Substitute the value of x back into the equation:
Now that we have found the value of x, we can substitute it back into one of the simplified equations to solve for y. Let's use Equation 1:

y = 16 / √x
y = 16 / √39.0625
y = 16 / 6.25
y = 2.56

Conclusion:
The values of x and y that satisfy the given equations are x = 39.0625 and y = 2.56, respectively.