How do you simplify sqrt(37-4sqrt(75)) ?

1 Answer
George C.
Jan 16, 2018

sqrt(37-4sqrt(75)) = 5-2sqrt(3)

Explanation:

I think you mean the positive square root of 37-4sqrt(75)

Note that:

37-4sqrt(75) = 37-4sqrt(5^2*3)

color(white)(37-4sqrt(75)) = 37-20sqrt(3)

color(white)(37-4sqrt(75)) = 5^2-2(5)(2sqrt(3))+3*2^2

color(white)(37-4sqrt(75)) = (5-2sqrt(3))^2

Also note that 2sqrt(3) ~~ 3.464 < 5, So 5-2sqrt(3) > 0.

So:

sqrt(37-4sqrt(75)) = 5-2sqrt(3)

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