Question

How do you simplify `3/(4+4sqrt5)`?

Answer 1

See a solution process below:

Explanation:

First, factor the denominator and rewrite the expression as:

`3/((4 * 1) + (4 * sqrt(5))) => 3/(4(1 + sqrt(5))) =>`

`3/4 xx 1/(1 + sqrt(5))`

Next, multiply this expression by `(1 - sqrt(5))/(1 - sqrt(5))` to eliminate the radical in the denominator while keeping the value of the expression the same because we are multiplying it by a form of `1`:

`3/4 xx 1/(1 + sqrt(5)) xx (1 - sqrt(5))/(1 - sqrt(5)) =>`

`3/4 xx (1/(1 + sqrt(5)) xx (1 - sqrt(5))/(1 - sqrt(5))) =>`

`3/4 xx (1 - sqrt(5))/(1 - sqrt(5) + sqrt(5) - (sqrt(5))^2) =>`

`3/4 xx (1 - sqrt(5))/(1 - 0 - 5) =>`

`3/4 xx (1 - sqrt(5))/-4 =>`

`3/4 xx -(1 - sqrt(5))/4`

We can now multiply the two terms giving:

`-(3(1 - sqrt(5)))/16`

Answer 2

Multiply both the numerator and denominator by the conjugate of 4 + 4`sqrt5`, which is 4 - 4`sqrt5`. Then simplify.

Explanation:

`3/(4+4sqrt5)` * `(4-4sqrt5)/(4-4sqrt5)`

Multiply these fractions together. In the denominator, you can see `(a+b)(a-b)`, which becomes `a^2 - b^2`.

= `(12 - 12sqrt5)/(16-80)`

= `(12 - 12sqrt5)/(-64)`

= `(3-3sqrt5)/-16`