If X=(2+√5)1⁄2 + (2-√5)1⁄2 and Y=(2+√5)1⁄2 - (2-√5)1⁄2 ,the evaluate x^2+ y^2

Answer 1

I assume those 1/2 's are exponents so your question is

x = √(2+√5) + √(2-√5)
y = √(2+√5) - √(2-√5)

Now clearly √(2-√5) is not a real number , so this must deal with imaginary numbers.

for ease of typing let a = √2+√5 , b = √2 - √5
(where b is an imaginary number)

so x = a+b
and y = a-b

x^2 + y^2

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

now a^2 = 2 + 2√10 + 5 = 7+2√10
and b^2 = 2 - 2√10 + 5 = 7 - 2√10

so
2a^2 + 2b^2
= 2(7+2√10) + 2(7-2√10)
= 14 + 4√10 + 14 - 4√10
= 28

Answer 2

looks like I left out the exponents in line
= (x+y)^ - 2xy
about 1/2 way down the solution

should have been:

= (x+y)^2 - 2xy

(no effect on the answer)