How do you simplify #(sqrt(15)+sqrt(35)+sqrt(21)+5)/(sqrt(3)+2sqrt(5)+sqrt(7))# to get #(sqrt(7)+sqrt(3))/2# ?
1 Answer
Sep 1, 2017
See explanation...
Explanation:
Given:
#(sqrt(15)+sqrt(35)+sqrt(21)+5)/(sqrt(3)+2sqrt(5)+sqrt(7))#
We could rationalise the denominator by multiplying both numerator and denominator by:
#(sqrt(3)+2sqrt(5)-sqrt(7))(sqrt(3)-2sqrt(5)+sqrt(7))(sqrt(3)-2sqrt(5)-sqrt(7))#
...but since we have been given what is at least supposed to be the answer, we can work backwards instead...
Note that:
#(sqrt(7)+sqrt(3))(sqrt(3)+2sqrt(5)+sqrt(7))#
#=sqrt(7)(sqrt(3)+2sqrt(5)+sqrt(7))+sqrt(3)(sqrt(3)+2sqrt(5)+sqrt(7))#
#=sqrt(21)+2sqrt(35)+7+3+2sqrt(15)+sqrt(21)#
#=2(sqrt(15)+sqrt(35)+sqrt(21)+5)#
So, dividing both ends by
#(sqrt(15)+sqrt(35)+sqrt(21)+5)/(sqrt(3)+2sqrt(5)+sqrt(7)) = (sqrt(7)+sqrt(3))/2#