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

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7  
$\sqrt{40}=2\sqrt{10}$... –  J. M. Jul 23 '11 at 12:30
    
For more information,here: en.wikipedia.org/wiki/Nth_root –  Smart Man Jul 24 '11 at 1:42
    
Too trivial to be discussed here. –  Debashish Jun 19 at 9:25

2 Answers 2

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

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Observe that

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

Then

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

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