\((\sqrt{5+\sqrt5}-\sqrt{5-\sqrt5})^2=\)
A. \(10-4\sqrt5\)
B. \(10-2\sqrt5\)
C. \(20-8\sqrt5\)
D. \(20-4\sqrt5\)
E. \(20-2\sqrt5\)
Answer: A
Source: Magoosh
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\((\sqrt{5+\sqrt5}-\sqrt{5-\sqrt5})^2=\)
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Solution:
Let A = √(5 + √5) and B = √(5 - √5) and recall that (A - B)^2 = A^2 - 2AB + B^2. So here, we have:
A^2 = 5 + √5
B^2 = 5 - √5
and
AB = √(5 + √5) * √(5 - √5) = √(25 - 5) = √20 = 2√5
Therefore, we have:
(A - B)^2 = A^2 - 2AB + B^2 = 5 + √5 - 2(2√5) + 5 - √5 = 10 - 4√5
Answer: A
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