Shed those radicals

Algebra Level 2

( 14 − 16 − 4 7 ) 2 = ? \Large \left(\sqrt {14}-\sqrt {16-4\sqrt{7}}\right)^2 = \ ?


The answer is 2.

Rohit Udaiwal by Rohit Udaiwal , 20, India

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3 solutions

Kaniji Max
Nov 15, 2015

i have another solution.

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How do you get the fourth step?

Sergio Alejandro Acelas Avila - 5 years, 7 months ago

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Square the equation of squartroot 14 - squareroot 2, the fourth step and you will get the equation at the third step.

Son Goku - 5 years, 7 months ago

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From (sqrt(a) - sqrt(b))^2 = a + b - 2sqrt(ab)

Kaniji Max - 5 years, 6 months ago
Rohit Udaiwal
Nov 12, 2015

( 14 − 16 − 4 7 ) 2 = 14 + 16 − 4 7 − 2 14 16 − 4 7 = 30 − 4 7 − 4 14 4 − 7 = 30 − 4 7 − 4 56 − 14 7 = 30 − 4 7 − 4 ( 7 − 7 ) 2 = 30 − 4 7 − 4 ( 7 − 7 ) = 30 − 4 7 − 28 + 4 7 = 2. \left (\sqrt {14 }-\sqrt {16-4\sqrt {7}}\right)^{2}=14+16-4\sqrt {7}-2\sqrt{14}\sqrt {16-4\sqrt {7}} \\ =30-4\sqrt {7}-4\sqrt {14}\sqrt {4-\sqrt {7}} \\ =30-4\sqrt {7}-4\sqrt {56- 14\sqrt {7}} \\ =30-4\sqrt {7}-4\sqrt {\left (7-\sqrt {7}\right)^{2}} \\ =30-4\sqrt{7}-4 (7-\sqrt {7}) \\ =30-4\sqrt {7}-28+4\sqrt {7}\\ =2.

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Same solution!

Ryoha Mitsuya - 5 years, 7 months ago
Satyam Bhardwaj
Nov 16, 2015

I have a better solution, but don't have the time to write it out. Basically just take 4 common out from the root then take 2 \sqrt{2} common out from the square, so that it becomes a 2. The thing inside the square will then easily resolve to a 1, so the answer is 2.

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