Question

$if \: the \: value \: of \: x + {x}^{ - 1} = 3 \: \\ then \: find \: the \: value \: of \: {x}^{5} + {x}^{ - 5}$
Pls anyone know then solve it


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Answer 1

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@ Abhyajha

Answer 2

$Given,\\ \\ x + x^{-1} = 3 \\ \\ x + \frac{1}{x} = 3 \\ \\ \text{making square of both sides} \\ \\ x^{2} + \frac{1}{x^{2} } + 2 = 9 \\ \\ x^{2} + \frac{1}{x^{2} } = 7.................(i) \\ \\ Again , \\ \\ x + \frac{1}{x} = 3 \\ \\ \text{making cube of both sides} \\ \\ x^{3} + \frac{1}{x^{3}} + 3( x + \frac{1}{x}) } = 27 \\ \\ x^{3} + \frac{1}{x^{3}} + 3*3 = 27 \\ \\ x^{3} + \frac{1}{x^{3}} = 27 - 9 \\ \\ x^{3} + \frac{1}{x^{3}} = 18 ............................(ii)$

$\text{multiplying equation (i) and (ii)} \\ \\ \\ ( x^{2} + \frac{1}{x^{2}}) (x^{3} + \frac{1}{x^{3} }) = 18* 7 \\ \\ x^{5} + \frac{1}{x} + x + \frac{1}{x^{5}} = 126 \\ \\ x^{5} + \frac{1}{x^{5}} + (x + \frac{1}{x} ) = 126 \\ \\ x^{5} + \frac{1}{x^{5}} + 3 = 126 \\ \\ x^{5} + \frac{1}{x^{5}} = 123$