Find the integers 𝑎 and 𝑏 such that the fraction root 12 minus two over one plus
root three is equal to 𝑎 plus 𝑏 root three.
This fraction currently has a surd in the denominator. And being asked to write it as 𝑎 plus 𝑏 root three means we’re being asked to
rationalize the denominator. To rationalize the denominator, we change the sign between the integer and the surd
and we multiply both the numerator and the denominator by this value. This doesn’t change the original fraction as the fraction we’re multiplying by is
equal to one.
We need to expand the brackets in both the numerator and denominator of this
fraction. In the numerator, we have root 12 minus root 12 multiplied by root three minus two
plus two root three. And in the denominator, we have one plus root three minus root three minus root three
multiplied by root three. In the denominator, the positive root three and the negative root three cancel each
other out. And root three multiplied by root three is three, which means there will be no surds
involved in the denominator.
In the numerator, root 12 can be written as the square root of three multiplied by
four, which simplifies to two root three. Therefore, this fraction simplifies to two root three minus two multiplied by root
three multiplied by root three minus two plus two root three all over one minus
Now, there’s further simplification that we can do here. Two root three plus two root three is equal to four root. Three root three multiplied by root three is equal to three. So the second term becomes negative two multiplied by three, which is negative
six. Overall, we have four root three minus six minus two over negative two. This gives four root three minus eight over negative two. And the final step is to divide both terms in the numerator by negative two. This gives negative two root three plus four.
The value of 𝑎 is the integer part of this — so it’s four — and the value of 𝑏 is
the multiple of root three — so it’s negative two.