Question
If x = (√5-√3) / (√5+√3) and y
is reciprocal of x. then what is the value of √(x3 + y3)?Solution
x = (√5 – √3)/(√5 + √3) y = 1/x ⇒ (√5 + √3)/(√5 – √3) (x + y) = [{(√5 – √3)/(√5 + √3)} + {(√5 + √3)/(√5 – √3)}] ⇒ (x + y) = [(√5 – √3)2 + (√5 + √3)2]/[(√5 – √3)(√5 + √3)] ⇒ (x + y) = [5 + 3 – 2 √15 + 5 + 3 + 2√15]/[(√5)2 – (√3)2] ⇒ (x + y) = 16/(5 – 3) = 16/2 = 8 (x + y)3 = x3 + y3 + 3 × x × y × (x + y) ⇒ x3 + y3 = (x + y)3 - 3 × x × (1/x) × (x + y) ⇒ 83 - 3 × 8 ⇒ 512 – 24 ⇒ 488 ∴ The value of (x3 + y3) is 488.
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