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Can $\frac{4+\sqrt{40}}{2}$ be simplified to $2+\sqrt{10}$ manually?

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$\sqrt{40}=2\sqrt{10}$... – J. M. Jul 23 '11 at 12:30
For more information,here: – Smart Man Jul 24 '11 at 1:42
Too trivial to be discussed here. – Debashish Jun 19 '14 at 9:25

$$\frac{4+\sqrt{40}}{2} = \frac{4+\sqrt{4\times 10}}{2} =\frac{4+\sqrt{4}\times\sqrt{10}}{2} = \frac{4+2\sqrt{10}}{2} = 2+\sqrt{10}$$

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Observe that

  1. $\dfrac{A+B}{C}=\dfrac{A}{C}+\dfrac{B}{C}$,
  2. $2=\sqrt{4}$,
  3. and $\dfrac{\sqrt{a}}{\sqrt{b}}=\sqrt{\dfrac{a}{b} }$.


$$\frac{4+\sqrt{40}}{2} = \frac{4}{2}+\frac{\sqrt{40}}{2} =2+\frac{\sqrt{40}}{\sqrt{4}}=2+\sqrt{\frac{40}{4}} = 2+\sqrt{10}.$$

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