# Finding $\sum\limits_{n=1}^{9999} \frac{1}{(\sqrt{n+1}+\sqrt{n})(\sqrt[4]{n}+\sqrt[4]{n+1})}$

How we can find

$$\sum_{n=1}^{9999} \frac{1}{(\sqrt{n+1}+\sqrt{n})(\sqrt[4]{n}+\sqrt[4]{n+1})}$$

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Hint: Compute $(\sqrt{n+1}+\sqrt{n})(\sqrt[4]{n+1}+\sqrt[4]{n})(\sqrt[4]{n+1}-\sqrt[4]{n})$.
• Warm up: Simplify the expression $(a^2+b^2)(a+b)(a-b)$.
• Pre-warm up: Simplify the expression $(a+b)(a-b)$ and deduce that $\sum\limits_{n=1}^{99}\frac1{\sqrt{n+1}+\sqrt{n}}=9$.
Great. Also applies to $$\sum_{n=1}^{99999999}\frac1{(\sqrt{n+1}+\sqrt n)(\sqrt[4]{n+1}+\sqrt[4] n)(\sqrt[8]{n+1}+\sqrt[8] n)}.$$ –  Hagen von Eitzen Sep 15 '12 at 16:32
I don't think he gets the hint. What happens if you multiply top and bottom by $\sqrt[4]{n+1}-\sqrt[4]n$? –  Mike Sep 15 '12 at 17:22