If x + (1/x) = √13, then what is the value of x 5- (1/x 5)?
393
Step 1 — Square the given:
\[\left(x+\tfrac{1}{x}\right)^2 = 13 \implies x^2+\tfrac{1}{x^2} = 11\]
Step 2 — Find \(x-\tfrac{1}{x}\):
\[\left(x-\tfrac{1}{x}\right)^2 = \left(x+\tfrac{1}{x}\right)^2 - 4 = 9 \implies x-\tfrac{1}{x} = 3\]
Step 3 — Find \(x^3-\tfrac{1}{x^3}\):
\[\left(x-\tfrac{1}{x}\right)^3 = x^3-\tfrac{1}{x^3} - 3\left(x-\tfrac{1}{x}\right)\]
\[27 = x^3-\tfrac{1}{x^3} - 9 \implies x^3-\tfrac{1}{x^3}=36\]
Step 4 — Combine: Using \(\left(x^3-\tfrac{1}{x^3}\right)\left(x^2+\tfrac{1}{x^2}\right) = \left(x^5-\tfrac{1}{x^5}\right)+\left(x-\tfrac{1}{x}\right)\):
\[x^5-\tfrac{1}{x^5} = 36 \times 11 - 3 = 393\]
The value is 393.
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