Simplifying Square Roots

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10 + 4 6 + 10 − 4 6 \sqrt{10+4\sqrt{6}}+\sqrt{10-4\sqrt{6}}

Simplify the above square root as much as you can.

8 6 8\sqrt{6} 4 4 2 10 2\sqrt{10} Cannot be simplified any more 20 20 10 + 96 + 10 − 96 \sqrt{10+\sqrt{96}}+\sqrt{10-\sqrt{96}}

Terry Yu by Terry Yu , 17, USA

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1 solution

Terry Yu
Apr 24, 2017

10 + 4 6 + 10 − 4 6 \sqrt{10+4\sqrt{6}}+\sqrt{10-4\sqrt{6}}

= 4 + 4 6 + 6 + 4 − 4 6 + 6 =\sqrt{4+4\sqrt{6}+6}+\sqrt{4-4\sqrt{6}+6}

= ( 2 + 6 ) 2 + ( 2 − 6 ) 2 =\sqrt{(2+\sqrt{6})^2}+\sqrt{(2-\sqrt{6})^2}

= ( 2 + 6 ) 2 + ( 2 − 6 ) 2 =(\sqrt{2+\sqrt{6}})^2+(\sqrt{2-\sqrt{6}})^2

= 2 + 6 + 2 − 6 =2+\sqrt{6}+2-\sqrt{6}

= 2 + 2 = 4 =2+2=\boxed{4}

The way to do these problems is to write the inside into a square, then cancel out the squares and the square roots to get a answer.

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