Roots problem

ok lets solve it algebraically

a = 9 , b = sqrt(80)

X =sqrt( a + b) , Y = sqrt(a -b )

thus the equation becomes

(X + Y )2 = x2 + y2 + 2xy = (a + b) + (a-b) + 2xy = 2a + 2sqrt { (a +b)(a-b) }

= 18 + 2 sqrt( a2 - b2) = 18 + 2 sqrt( 81 - 80 ) = 20

Thus 20 should be the answer

((√(9+√80)) + (√(9+√80)))^2 I have put a extra bracket before 9 to state the problem as in the Q.

For simplicity Let 9 = a and 80 = b

((√(a+√b)) + (√(a-√b)))^2
a + √b + a -√b +2(√(a+√b)(√a-√b)) ----- (x+y )^2 = x^2 + y^2 + 2xy
2a + 2 (√(a^2 - b)) ----- (x+y) (x-y) = x^2 - y^2
2*9 + 2 (√(81-80) ----- Substitute a and b with actual nos.
18 + 2

The problem is incomplete without the answer options as i am not able to check whether I am making a mistake.

Thank you both of you for your responses. That helped a lot.

If the answer choices were spread out, the fastest way to solve this would be via estimation.

First, let's recognize that root80 is very close to 9. If we let root80 = 9, we get:

((root(9+9)) +(root(9-9)))^2 = (root18)^2 = 18

So, if there were only one answer choice close to 18, we'd just pick it and be done.