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If
x=(1)/(sqrt2+sqrt3), find the value of
x^(2)+(1)/(x^(2))

If $x=\frac{1}{\sqrt{2}+\sqrt{3}}$ , find the value of $x^{2}+\frac{1}{x^{2}}$

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Find the vertex of the transformed function

Full solution

Q. If

x=\frac{1}{\sqrt{2}+\sqrt{3}}

, find the value of

x^{2}+\frac{1}{x^{2}}

Find $x^2$ : First, let's find the value of $x^2$ . $x = \frac{1}{(\sqrt{2} + \sqrt{3})}$ Square both sides to get $x^2$ . $x^2 = \frac{1^2}{((\sqrt{2} + \sqrt{3})^2)}$
Simplify $x^2$ : Now, let's simplify $\left(1^2\right)/\left(\left(\sqrt{2} + \sqrt{3}\right)^2\right)$ . $x^2 = 1/\left(2 + 2\sqrt{2}\sqrt{3} + 3\right)$ $x^2 = 1/\left(5 + 2\sqrt{6}\right)$
Find $1/x^2$ : Next, we need to find the value of $1/x^2$ . $1/x^2 = (5 + 2\sqrt{6})/1$
Add $x^2$ and $\frac{1}{x^2}$ : Now, let's add $x^2$ and $\frac{1}{x^2}$ . $x^2 + \frac{1}{x^2} = \frac{1}{5 + 2\sqrt{6}} + \frac{5 + 2\sqrt{6}}{1}$
Common denominator for addition: To add these fractions, they need a common denominator. $\newline$ $x^2 + \frac{1}{x^2} = \frac{1 + (5 + 2\sqrt{6})^2}{5 + 2\sqrt{6}}$

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