How do you rationalize the denominator and simplify 1/(sqrt3 - sqrt5 - 2)?
1 Answer
1/(sqrt(3)-sqrt(5)-2) = -10/47-4/47sqrt(15)+7/47sqrt(3)-1/47sqrt(5)
Explanation:
Multiply numerator and denominator by:
(sqrt(3)+sqrt(5)-2)(sqrt(3)-sqrt(5)+2)(sqrt(3)+sqrt(5)+2)
First note that:
(sqrt(3)-sqrt(5)+2)(sqrt(3)+sqrt(5)+2)
=((sqrt(3)+2)-sqrt(5))((sqrt(3)+2)+sqrt(5))
=(sqrt(3)+2)^2-(sqrt(5))^2
=2+4sqrt(3)+4-5
= 1+4sqrt(3)
Similarly:
(sqrt(3)-sqrt(5)-2)(sqrt(3)+sqrt(5)-2)
=((sqrt(3)-2)-sqrt(5))((sqrt(3)-2)+sqrt(5))
=(sqrt(3)-2)^2-(sqrt(5))^2
=2-4sqrt(3)+4-5
= 1-4sqrt(3)
So the denominator becomes:
(1-4sqrt(3))(1+4sqrt(3))=1^2-(4sqrt(3))^2=1-48=-47
Meanwhile, the numerator becomes:
(sqrt(3)+sqrt(5)-2)(1+4sqrt(3))
=sqrt(3)+sqrt(5)-2 + 4sqrt(3)(sqrt(3)+sqrt(5)-2)
=sqrt(3)+sqrt(5)-2+12+4sqrt(15)-8sqrt(3)
=10+4sqrt(15)-7sqrt(3)+sqrt(5)
So:
1/(sqrt(3)-sqrt(5)-2) = (10+4sqrt(15)-7sqrt(3)+sqrt(5))/(-47)
=-10/47-4/47sqrt(15)+7/47sqrt(3)-1/47sqrt(5)
