Carcass wrote:
\((\sqrt{9+\sqrt{80}}+\sqrt{9-\sqrt{80}})^2=\)
A. 1
B. \(9 - 4*\sqrt{5}\)
C. \(18 - 4*\sqrt{5}\)
D. 18
E. 20
Notice that the expression is in the form (
x +
y)², where
x = √(9 + √80) and
y = √(9 - √80)We know that (
x +
y)² =
x² + 2
xy +
y²
If
x = √(9 + √80), then x² = 9 + √80
If
y = √(9 - √80), then y² = 9 - √80
Finally,
xy = [
√(9 + √80)][
√(9 - √80)] = 81 - 80 = 1
So, we get:
(
x +
y)² =
x² + 2
xy +
y²
= (9 + √80) + 2(1) + (9 - √80)
= 9 + √80 + 2 + 9 - √80
= 20
Answer: E