If x + 1/x = 3, calculate x2

Question:

If $x+1 / x=3$, calculate $x^{2}+1 / x^{2}, x^{3}+1 / x^{3}, x^{4}+1 / x^{4}$

Solution:

Given, $x+1 / x=3$

We know that $(x+y)^{2}=x^{2}+y^{2}+2 x y$

$(x+1 / x)^{2}=x^{2}+1 / x^{2}+(2 * x * 1 / x)$

$3^{2}=x^{2}+1 / x^{2}+2$

$9-2=x^{2}+1 / x^{2}$

$x^{2}+1 / x^{2}=7$

Squaring on both sides

$\left(x^{2}+1 / x^{2}\right)^{2}=7^{2}$

$x^{4}+1 / x^{4}+2^{*} x^{2} * 1 / x^{2}=49$

$x^{4}+1 / x^{4}+2=49$

$x^{4}+1 / x^{4}=49-2$

$x^{4}+1 / x^{4}=47$

Again, cubing on both sides

$(x+1 / x)^{3}=3^{3}$

$x^{3}+1 / x^{3}+3 x^{*} 1 / x(x+1 / x)=27$

$x^{3}+1 / x^{3}+\left(3^{*} 3\right)=27$

$x^{3}+1 / x^{3}+9=27$

$x^{3}+1 / x^{3}=27-9$

$x^{3}+1 / x^{3}=18$

Hence, the values are $x^{2}+1 / x^{2}=7, x^{4}+1 / x^{4}=47, x^{3}+1 / x^{3}=18$

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