Question

If x+1/x = 9, find the value of x^4+1/x^4.
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Answer 1

If x + 1/x = 9 find the value of x⁴ + 1/x⁴

First Square both the sides

$\mathtt{ {(x + \frac{1}{x}) }^{2} } = {(9)}^{2} \\ \\ \implies\mathtt{{(x + \frac{1}{x}) }^{2} = 81} \\ \\ \implies\mathtt{ {x}^{2} + 2 \times x \times \frac{1}{x} + {( \frac{1}{x}) }^{2} = 81 } \\ \\ \implies\mathtt{ {x}^{2} + 2 + \frac{1}{ {x}^{2} } = 81 } \\ \\ \implies\mathtt{{x}^{2} + \frac{1}{ {x}^{2} } = 79}$

Now, square both the sides again

$\mathtt{ {( {x}^{2} + \frac{1}{2} )}^{2} = {(79)}^{2} }\\ \\ \implies\mathtt{{( {x}^{2} + \frac{1}{2} )}^{2} =6241}\\ \\ \implies\mathtt{ {x}^{4} + 2 + \frac{1}{ {x}^{4} } = 6241 } \\ \\ \implies\mathtt{ {x}^{4} + \frac{1}{ {x}^{4} } = 6239 \: ans }$