Can you write (a+b)^0.5 - (a-b)^0.5 as the square root of a difference?

2 Answers
George C.
Mar 4, 2018

(a+b)^0.5-(a-b)^0.5 = sqrt(2a-2sqrt(a^2-b^2))

Explanation:

If (a+b)^0.5 - (a-b)^0.5 a.k.a. sqrt(a+b)-sqrt(a-b) can be written as the square root of a difference, then squaring it should give us a difference...

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

color(white)((sqrt(a+b)-sqrt(a-b))^2) = (a+b)-2sqrt(a+b)sqrt(a-b)+(a-b)

color(white)((sqrt(a+b)-sqrt(a-b))^2) = 2a-2sqrt((a+b)(a-b))

color(white)((sqrt(a+b)-sqrt(a-b))^2) = 2a-2sqrt(a^2-b^2)

So yes, (a+b)^0.5 - (a-b)^0.5 can be written as the square root of the difference of 2a and 2sqrt(a^2-b^2), namely:

(a+b)^0.5-(a-b)^0.5 = sqrt(2a-2sqrt(a^2-b^2))

Cesareo R.
Mar 4, 2018

sqrt(2a-2sqrt(a^2-b^2))

Explanation:

sqrt(x-y) = sqrt(a+b)-sqrt(a-b)

squaring both sides

x-y = 2a-2sqrt(a^2-b^2) and then

sqrt(x-y)=sqrt(a+b)-sqrt(a-b) = sqrt(2a-2sqrt(a^2-b^2))

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