Question
Download Solution PDFIf
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
x - (1/x) = √6
Formula used:
x8 - (1/x8) = {x4 + (1/x4)} × {x2 + (1/x2)} × {x + (1/x)} × {x - (1/x)}
If x - (1/x) = a, then x + (1/x) = √(a2 + 4)
Calculation:
x - (1/x) = √6
x2 + (1/x2) = (√6)2 + 2 = 8
Square both sides:
x4 + (1/x4) = (8)2 - 2 = 62
Using, If x - (1/x) = a, then x + (1/x) = √{(√a)2 + 4} .
x + (1/x) = √{(√6)2 + 4} = √10
Substituting the values, we get
x8 - (1/x8) = {x4 + (1/x4)} × {x2 + (1/x2)} × {x + (1/x)} × {x - (1/x)}
⇒ 62 × 8 × √10 × √6 = 496 × 2 × √15 = 992√15
∴ The correct answer is 992√15.
Shortcut Trick
If a - 1/a = m,
• Then a2 +1/a2 = m2 + 2
• Then a4 + 1/a4 = m4 + 4m2 + 2
if a2 +1/a2 = m , then a +1/a = √(m+2)
⇒ x - 1/x =√6
⇒ x2+1/x2 = (√6)2 +2 = 8
⇒ x +1/x = √10
⇒ x4+ 1/x4 = (√6)4+4(√6)2 +2 = 36+24+2 = 62
⇒ x8 - (1/x8) = {x4 + (1/x4)} × {x2 + (1/x2)} × {x + (1/x)} × {x - (1/x)}
⇒ 62 × 8 × √10 × √6 = 496 × 2 × √15
⇒ 992√15
Last updated on Jul 19, 2025
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