Question
Download Solution PDFIf x > 0, and \(x^4 + \frac{1}{x^4} = 254\), what is the value of \(x^5 + \frac{1}{x^5}\)?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
\(x^4 + \frac{1}{x^4} = 254\)
Formula used:
\(x^5 + \frac{1}{x^5} \) = ( \(x^3 + \frac{1}{x^3} \) ) × ( \(x^2 + \frac{1}{x^2} \)) - \((x + \frac{1}{x} )\)
Calculation:
\(x^4 + \frac{1}{x^4} +2 = [{x^2 + \frac{1}{x^2}}]^2\)
⇒ \( [{x^2 + \frac{1}{x^2}}]^2\) = 254 + 2
⇒ \( [{x^2 + \frac{1}{x^2}}]\) = 16
⇒ \(x^2 + \frac{1}{x^2} +2 = [{x + \frac{1}{x}}]^2 \)
⇒ \((x + \frac{1}{x} )\) = √16+2
⇒ \((x + \frac{1}{x} )\) = 3√2
⇒ \((x^3 + \frac{1}{x^3} )\) = (3√2)3 - 3 × 3√2
⇒ \((x^3 + \frac{1}{x^3} )\) = 54√2 - 9√2
⇒ \((x^3 + \frac{1}{x^3} )\) = 45√2
⇒\(x^5 + \frac{1}{x^5} \) = 45√2 × 16 - 3√2
⇒ \(x^5 + \frac{1}{x^5} \) = 717√2
Last updated on Jul 16, 2025
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