If x = 1 + √2 and y = 1 - √2, find the value of (x² + y²)².

Question: If x = 1 + √2 and y = 1 - √2, find the value of (x² + y²)².

Solution:
Given that, x = 1 + √2 and y = 1 - √2
∴ x + y = 1 + √2 + 1 - √2
= 2

And, xy = (1 + √2)(1 - √2)
= 1² - (√2)²
= 1 - 2
= - 1

Now,
x² + y² = (x + y)² - 2xy
= (2)² - 2(- 1)
= 4 + 2
= 6

∴ (x² + y²)² = 6²
= 36