Math Book Cambodia: If `x+1/x=2` Find the value of `x^5+1/x^5`

Sunday, September 11, 2022

If $x + \frac{1}{x} = 2$ $x+1/x=2$ Find the value of $x^{5} + \frac{1}{x^{5}}$ $x^5+1/x^5$

As we had: $x + \frac{1}{x} = 2$ $x+1/x=2$ $\to {(x + \frac{1}{x})}^{2} = 4$ $rightarrow(x+1/x)^2=4$ $\leftrightarrow x^{2} + \frac{1}{x^{2}} = 2$ $leftrightarrowx^2+1/x^2=2$

We continue with $(x^{2} + \frac{1}{x^{2}}) (x + \frac{1}{x}) = 4$ $(x^2+1/x^2)(x+1/x)=4$ $\leftrightarrow x^{3} + x + \frac{1}{x} + \frac{1}{x^{3}} = 4$ $leftrightarrowx^3+x+1/x+1/x^3=4$

$\leftrightarrow x^{3} + \frac{1}{x^{3}} = 2$ $leftrightarrowx^3+1/x^3=2$

We do that again: $(x^{3} + \frac{1}{x^{3}}) (x + \frac{1}{x}) = 4$ $(x^3+1/x^3)(x+1/x)=4$ $\leftrightarrow x^{5} + x + \frac{1}{x} + \frac{1}{x^{5}} = 4$ $leftrightarrowx^5+x+1/x+1/x^5=4$

Therefore, $x^{5} + \frac{1}{x^{5}} = 2$ $x^5+1/x^5=2$ Because $x + \frac{1}{x} = 2$ $x+1/x=2$

Solution By: Thin Sokkean

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Sunday, September 11, 2022

If $x + \frac{1}{x} = 2$ $x+1/x=2$ Find the value of $x^{5} + \frac{1}{x^{5}}$ $x^5+1/x^5$

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Sunday, September 11, 2022

If x+1x=2x+1x=2x+1/x=2 Find the value of x5+1x5x5+1x5x^5+1/x^5

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If $x + \frac{1}{x} = 2$ $x+1/x=2$ Find the value of $x^{5} + \frac{1}{x^{5}}$ $x^5+1/x^5$