simplify: sqrt(6)/(4+sqrt(2))?

Answer 1

One thing with fractions is that we NEVER want a radical in the denominator. To get rid of it, we multiply the numerator and denomiantor by the denominator's conjugate.

The conjguate of 4 + sqrt(2) is 4 - sqrt(2), and note that multiplying a binomial (a + b) by its conjugate (a - b) yields a difference of squares, a^2 - b^2.

sqrt(6) / [4 + sqrt(2)]

Multiply top and bottom by [4 - sqrt(2)]

sqrt(6) [4 - sqrt(2)] / { [4 + sqrt(2)][4 - sqrt(2)] }

The bottom becomes a difference of squares.

sqrt(6) [4 - sqrt(2)] / { 4^2 - [sqrt(2)]^2 }

The square of the square root of 2 is 2.

sqrt(6) [4 - sqrt(2)] / { 16 - 2 }

Expand the top, simplify the bottom

[4sqrt(6) - sqrt(6)sqrt(2)] / 14

The multiplication of two radicals, sqrt(x) * sqrt(y) is sqrt(xy), so

[4sqrt(6) - sqrt(12)] / 14

The square root of 12 is reducible, since sqrt(12) = sqrt(4*3), which is equal to 2sqrt(3).

[4sqrt(6) - 2sqrt(3)] / 14

Factor out a 2,

2[2sqrt(6) - sqrt(3)] / 14

The 2 on the top cancels from the 14 on the bottom. The 14 on the bottom becomes 7.

[2sqrt(6) - sqrt(3)] / 7

Answer 2

Simplify (√6)/(4 + √2).

We want to get the radical out of the denominator. To do that we will multiply both numerator and denominator by the conjugate
4 - √2

(√6) / (4 + √2) = (√6)(4 - √2) / {(4 + √2)(4 - √2)}
= (4√6 - √12) / (16 - 2) = (4√6 - 2√3) / 14
= (2√6 - √3) / 7

Answer 3

(sqrt(6))/(4 + sqrt(2))

Multiply top and bottom by 4 - sqrt(2)

(sqrt(6)(4 - sqrt(2)))/(((4 + sqrt(2))(4 - sqrt(2)))
(4sqrt(6) - sqrt(12))/(16 - 4sqrt(2) + 4sqrt(2) - 2)
(4sqrt(6) - 2sqrt(3))/(14)

ANS : (2sqrt(6) - sqrt(3))/7