how does (2) / √[(2+√(2))] become (2-√(2))√[(2+√(2))]
steps too please
To understand how to simplify the expression (2) / √[(2+√(2))], let's break down the steps:
Step 1: Rationalizing the denominator
To simplify the expression, we need to get rid of the square root in the denominator. We achieve this by multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate of the denominator is (2-√(2)).
So, we multiply the numerator and denominator by (2-√(2)):
[(2) / √[(2+√(2))]] * [(2-√(2)) / (2-√(2))]
This results in:
(2 * (2-√(2))) / [(2+√(2)) * (2-√(2))]
Step 2: Simplifying the denominator
Multiplying the denominator using the distributive property, we get:
(2 * (2-√(2))) / [4 - 2√(2) + 2√(2) - (√(2))^2]
Simplifying further:
(2 * (2-√(2))) / [4 - (√(2))^2]
Step 3: Simplifying the denominator further
(√(2))^2 simplifies to 2:
(2 * (2-√(2))) / (4 - 2)
Step 4: Simplifying the denominator completely
4 - 2 = 2:
(2 * (2-√(2))) / 2
Simplifying further:
(2-√(2))
So, (2) / √[(2+√(2))] simplifies to (2-√(2)).