Question
Download Solution PDFIf x > 0 and \(\rm x^4 + \frac{1}{x^4} = 142\), what is the value of \(\rm x^7 + \frac{1}{x^7} \)?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven
\(\rm x^4 + \frac{1}{x^4} = 142\)
Formula used
(x + 1/x)2 = x2 + 1/x2 + 2
x3 + 1/x3 = (x + 1/x)3 - 3(x + 1/x)
Calculation
⇒ x4 + 1/x4 + 2 = (x2 + 1/x2)2
⇒ 142 + 2 = (x2 + 1/x2)2
⇒ (x2 + 1/x 2) = √144 = 12
⇒ (x + 1/x)2 = x2 + 1/x2 + 2
⇒ (x + 1/x)2 = 12 + 2
⇒ (x + 1/x) = √14
⇒ x3 + 1/x3 = (x + 1/x)3 - 3(x + 1/x)
= (√14)3 - 3(√14)
= 14√14 - 3√14 = 11√14
(x7 + 1/x7) = (x4 + 1/x4) × (x3 + 1/x3) - (x + 1/x)
= 142 × 11√14 - √14
= 1562√14 - √14
= √14 (1562 - 1)
= 1561√14
The answer is 1561√14
Last updated on Jun 13, 2025
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