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How can we simplify the equation

$$\sqrt{5+\sqrt{3}}+\sqrt{5-\sqrt{3}}$$ How can we represent it as one radical?

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4 Answers 4

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Let $u=\sqrt{5+\sqrt{3}}$, $v=\sqrt{5-\sqrt{3}}$.

Then $u^2+v^2=10$ and $uv=\sqrt{22}$.

Therefore, $(u+v)^2=u^2+v^2+2uv=10+2\sqrt{22}$ and so $u+v=\sqrt{10+2\sqrt{22}}$.

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$$x=\sqrt{5+\sqrt3}+\sqrt{5-\sqrt3}$$ $$x^2=10+2\sqrt{22}$$ $$x=\sqrt{10+2\sqrt{22}}$$

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HINT:

$$\sqrt{\frac{A + \sqrt{A^2-B}}{2}} \pm \sqrt{\frac{A - \sqrt{A^2-B}}{2}} = \sqrt{A \pm \sqrt{B}}$$

Now just find the suitable values for $A,B$

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Square the expression to get

$5 + \sqrt{3} + 5 - \sqrt{3} + 2\sqrt{(5 + \sqrt{3}) (5 - \sqrt{3})}$

and simplify.

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