Question 617254
{{{(3sqrt(5))/(2+2sqrt(5))}}}
First let's rationalize the denominator. Rationalizing two-term denominators with square roots takes advantage of the {{{(a+b)(a-b) = a^2-b^2}}} pattern. The right side, as you can see is a difference of perfect squares. The left side is a product of two-term expressions. The pattern shows us how to take a two-term expression, an (a+b) or (a-b), and turn it into an expression of perfect squares.<br>
Your denominator is a two-term sum. IOW, an (a+b). To turn it into an expression of perfect squares we need to multiply it by its corresponding (a-b):
{{{((3sqrt(5))/(2+2sqrt(5)))((2-2sqrt(5))/(2-2sqrt(5)))}}}
Multiplying the denominators is easy; just use the pattern. To multiply the numerators we will need to use the distributive property:
{{{(3sqrt(5)*2-3sqrt(5)*2sqrt(5))/((2)^2-(2sqrt(5))^2)}}}
which simplifies as follows:
{{{(6sqrt(5)-6*5)/(4-(4*5))}}}
{{{(6sqrt(5)-30)/(4-(20))}}}
{{{(6sqrt(5)-30)/(-16)}}}
Now we can split the fraction to get the {{{a*sqrt(5)-b}}} form:
{{{(6sqrt(5))/(-16)-(30)/(-16)}}}
which simplifies to:
{{{(-3/8)sqrt(5)-((-15)/8)}}}
So {{{a = (-3)/8}}} and {{{b = -15/8}}}