For positive x and y, which expression is equivalent to (x^1/2-y^1/2)^2?
a. x-y
b. x+2sqrt(xy)+y
c. x-2sqrt(xy)+y
I need all of the steps to get to the right answer.
Thanks
(√x - √y)^2
= x - 2√(xy) + y
To simplify the expression (x^1/2 - y^1/2)^2, we can use the properties of exponents and expand the expression.
Step 1: Rewrite the exponents as fractional exponents.
(x^1/2 - y^1/2)^2 = (sqrt(x) - sqrt(y))^2
Step 2: Expand the expression using the pattern (a - b)^2 = a^2 - 2ab + b^2.
(x^1/2 - y^1/2)^2 = (sqrt(x))^2 - 2(sqrt(x))(sqrt(y)) + (sqrt(y))^2
Step 3: Simplify each term.
(sqrt(x))^2 = x
(sqrt(y))^2 = y
2(sqrt(x))(sqrt(y)) = 2√(xy)
Therefore:
(x^1/2 - y^1/2)^2 = x - 2√(xy) + y
So, the correct expression equivalent to (x^1/2 - y^1/2)^2 is Option C: x - 2√(xy) + y.