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If x+1 x = 5 , then evaluate x⁴+1 x⁴

x + \frac{1}{x} = 5 \Rightarrow {(x + \frac{1}{x})}^{2} = 5^{2} {squaring both sides} \Rightarrow x^{2} + \frac{1}{x^{2}} + 2 \times x \times \frac{1}{x} = 25 \Rightarrow x^{2} + \frac{1}{x^{2}} + 2 = 25 \Rightarrow x^{2} + \frac{1}{x^{2}} = 23 \Rightarrow {(x^{2} + \frac{1}{x^{2}})}^{2} = {(23)}^{2} {again squaring both sides} \Rightarrow x^{4} + \frac{1}{x^{4}} + 2 \times x^{2} \times \frac{1}{x^{2}} = 529 \Rightarrow x^{4} + \frac{1}{x^{4}} + 2 = 529 \Rightarrow x^{4} + \frac{1}{x^{4}} = 527

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