Problem:
Find the extreme values of √(x^2y^2) when 13x^2 - 10xy + 13y^2 = 72.

Solution:

Step 1: Simplify the given equation
Start by simplifying the given equation 13x^2 - 10xy + 13y^2 = 72. Notice that this equation is similar to the standard form of an ellipse equation, which is (x^2/a^2) + (y^2/b^2) = 1. Therefore, we can rewrite the equation as:

13x^2 - 10xy + 13y^2 = 72
Divide both sides by 72 to make the equation equal to 1:
(x^2/72) - (10xy/72) + (y^2/72) = 1

Step 2: Rewrite the equation in terms of a and b
Compare the rewritten equation with the standard form of an ellipse equation, which is (x^2/a^2) + (y^2/b^2) = 1. By comparing the two equations, we can determine the values of a and b:

a^2 = 72
b^2 = 72

Step 3: Find the extreme values of √(x^2y^2)
To find the extreme values of √(x^2y^2), we can use the fact that the square root function is a strictly increasing function. This means that the maximum and minimum values of √(x^2y^2) will occur at the maximum and minimum values of x^2y^2.

Since x^2y^2 is always non-negative, the maximum and minimum values will occur when x^2y^2 is equal to the maximum and minimum values of x^2y^2 respectively.

Step 4: Find the maximum and minimum values of x^2y^2
Substitute the values of a and b into the equation x^2y^2:

x^2y^2 = (a^2)(b^2)
= (72)(72)
= 5184

Therefore, the maximum and minimum values of x^2y^2 are 5184.

Step 5: Find the extreme values of √(x^2y^2)
Now, substitute the maximum and minimum values of x^2y^2 into the function √(x^2y^2):

√(x^2y^2) = √(5184)
= 72

Therefore, the extreme values of √(x^2y^2) are 72.

Conclusion:
The extreme values of √(x^2y^2) when 13x^2 - 10xy + 13y^2 = 72 are 72.