Answer:
-3/2 + 1/2 sqrt(5)
Step-by-step explanation:
sqrt(2) - sqrt(10)
-------------------------
sqrt(2) + sqrt(10)
To clear the square root in the denominator, we multiply by the conjugate
sqrt(2) - sqrt(10). What we do to the bottom, we do to the top
sqrt(2) - sqrt(10) sqrt(2) - sqrt(10)
------------------------- * -------------------------------
sqrt(2) + sqrt(10) sqrt(2) - sqrt(10)
sqrt(2) * sqrt(2) -sqrt(2)sqrt(10) -sqrt(2) sqrt(10) +sqrt(10)sqrt(10)
-------------------------------------------------------------------------------------------
sqrt(2) * sqrt(2) -sqrt(2)sqrt(10) +sqrt(2) sqrt(10) - sqrt(10)sqrt(10)
Combining like terms
sqrt(2) * sqrt(2) -2sqrt(2)sqrt(10) -sqrt(2) sqrt(10) + sqrt(10)sqrt(10)
-------------------------------------------------------------------------------------------
sqrt(2) * sqrt(2) - sqrt(10)sqrt(10)
2 - 2 sqrt(20) +10
-------------------------
2-10
12 - 2 sqrt(20)
--------------------------
-8
sqrt(20) = sqrt(4) sqrt(5) = 2sqrt(5)
Replacing sqrt(20)
12 - 2 *2 sqrt(5)
--------------------------
-8
12 -4 sqrt(5)
---------------------
-8
Factor out a 4
4(3 -sqrt(5))
---------------------
4 (-2)
Canceling the 4
3-sqrt(5)
-----------------
-2
Breaking the fraction into 2 pieces
-3/2 + 1/2 sqrt(5)
