SOLUTION: Rationalize the numerator for the following expression. (a) {{{ (1 - sqrt(3))/3 }}} (b) {{{ (sqrt(p)+sqrt(p^2-1))/(sqrt(p)-sqrt(p^2-1)) }}}

Algebra ->  Radicals -> SOLUTION: Rationalize the numerator for the following expression. (a) {{{ (1 - sqrt(3))/3 }}} (b) {{{ (sqrt(p)+sqrt(p^2-1))/(sqrt(p)-sqrt(p^2-1)) }}}      Log On


   

Question 718131: Rationalize the numerator for the following expression.
(a) +%281+-+sqrt%283%29%29%2F3+
(b) +%28sqrt%28p%29%2Bsqrt%28p%5E2-1%29%29%2F%28sqrt%28p%29-sqrt%28p%5E2-1%29%29+
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
a) 3 is rational
b) Multiply the numerator and denominator by sqrt%28p%29%2Bsqrt%28p%5E2-1%29. This will be easier if you think of sqrt%28p%29 as "a" and sqrt%28p%5E2-1%29 as "b" and use the %28a%2Bb%29%5E2+=+a%5E2%2B2ab%2Bb%5E2 pattern to multiply the numerators and the %28a-b%29%28a%2Bb%29+=+a%5E2-b%5E2 pattern to multiply the denominators. (The second pattern, in fact, is how you figure out that multiplying by sqrt%28p%29%2Bsqrt%28p%5E2-1%29 will help. We already have a ("a"-"b"). If we multiply by the corresponding ("a"+"b") we will get and expression of only perfect squares, making the square roots disappear!)