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

2 Answers
Jul 3, 2017

Answer:

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#

Jul 3, 2017

Answer:

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#