How do you simplify #(sqrt 6 - sqrt 5) / (sqrt 6 + sqrt 5)#?
1 Answer
Explanation:
Your goal here is to rationalize the denominator by multiplying it by its conjugate.
The conjugate of a binomial can be determined by changing the sign of the second term. In your case, you would have
#sqrt(6) + sqrt(5) -> underbrace(sqrt(6) color(red)(-) sqrt(5))_(color(blue)("conjugate"))#
So, multiply the fraction by
#(sqrt(6) - sqrt(5))/(sqrt(6) + sqrt(5)) * (sqrt(6) - sqrt(5))/(sqrt(6) - sqrt(5)) = ((sqrt(6) - sqrt(5))(sqrt(6) - sqrt(5)))/((sqrt(6) + sqrt(5))(sqrt(6) - sqrt(5))#
The denominator takes the form
#color(blue)( (a-b)(a+b) = a^2 - b^2)#
and can thus be written as
#(sqrt(6) + sqrt(5))(sqrt(6) - sqrt(5)) = (sqrt(6))^2 - (sqrt(5))^2 = 6 - 5 = 1#
The numerator takes the form
#color(blue)( (a-b)^2 = a^2 - 2ab + b^2)#
and can be written as
#(sqrt(6) - sqrt(5))(sqrt(6) - sqrt(5)) = (sqrt(6))^2 - 2sqrt(30) + (sqrt(5))^2#
#=6 - 2sqrt(30) + 5#
#= 11 - 2sqrt(30)#
The expression becomes
#((sqrt(6) - sqrt(5))(sqrt(6) - sqrt(5)))/((sqrt(6) + sqrt(5))(sqrt(6) - sqrt(5))) = (11 - 2sqrt(30))/1 = color(green)(11 - 2sqrt(30))#
