Simplify (7+sqrt(5))/(sqrt(5)-1), leaving the answer in the form a+b*sqrt(5)

Step 1 - Identify the difference between the required form of the answer and the expression to be simplified: 

It can be seen that in order to simplify the expression we need to somehow get rid of the denominator.

Step 2 - Attempt a method to get rid of the denominator: 

The best way to get rid of the denominator is to multiply it by (sqrt(5)+1). In order to keep the expression the same we must do the same to the numerator as well. 

This gives us,


Step 3 - Carry out the necessary calculations:

The top parts of the fractions multiply together and the bottom parts of the fractions multiply together.

For the top part this gives us,

(7+sqrt(5))*(sqrt(5)+1) = 7*sqrt(5)+7+5+sqrt(5)

= 8*sqrt(5)+12

For the bottom part this gives us,

(sqrt(5)-1)*(sqrt(5)+1) = 5+sqrt(5)-sqrt(5)-1 = 4

Dividing the top part by the bottom part gives us,


Step 4 - Check that the answer is in the correct form.

The question asks for an answer in the form,


Our answer can be written in this format with a = 3 and b = 2.

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