How do you simplify sqrt3/(sqrt6 -1) - sqrt3/(sqrt6 + 1)√3√6−1−√3√6+1?

1 Answer
Lotusbluete
Feb 8, 2016

(2 sqrt(3)) / 52√35

Explanation:

You need to find the least common denominator to be able to subtract the two fractions.

In your case, the least common denominator is (sqrt(6) - 1)(sqrt(6) + 1)(√6−1)(√6+1), so you need to expand the first fraction with (sqrt(6) + 1)(√6+1) and the second one with (sqrt(6) - 1)(√6−1).

sqrt(3) / (sqrt(6) - 1) - sqrt(3) / (sqrt(6) + 1) = (sqrt(3)color(blue)((sqrt(6) + 1))) / ((sqrt(6) - 1)color(blue)((sqrt(6) + 1))) - (sqrt(3)color(green)((sqrt(6) - 1))) / ((sqrt(6) + 1)color(green)((sqrt(6) - 1))) √3√6−1−√3√6+1=√3(√6+1)(√6−1)(√6+1)−√3(√6−1)(√6+1)(√6−1)

= (sqrt(3)(sqrt(6) + 1) - sqrt(3)(sqrt(6) - 1)) / ((sqrt(6) - 1)(sqrt(6) + 1))=√3(√6+1)−√3(√6−1)(√6−1)(√6+1)

... use the formula (a + b)(a - b) = a^2 - b^2(a+b)(a−b)=a2−b2 to simplify the denominator...

= (sqrt(3) * sqrt(6) + sqrt(3) - sqrt(3) * sqrt(6) + sqrt(3)) / ((sqrt(6))^2 - 1^2)=√3⋅√6+√3−√3⋅√6+√3(√6)2−12

= (cancel(sqrt(3) * sqrt(6)) + sqrt(3) - cancel(sqrt(3) * sqrt(6)) + sqrt(3)) / (6 - 1)

= (2 sqrt(3)) / 5

= 2/5 sqrt(3)

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