Rationalizing a denominator.

Question

The question instructs to rationalize the denominator in the following fraction:

My solution is as follows:

The book's solution is

which is exactly the numerator in my solution.

Can someone confirm my solution or point what what I'm doing wrong?

Thank you.

Answer 1

Going from the first to the second line of your working, you seem to have said $$(\sqrt{6} -2)(\sqrt{6}+2) = 6+4$$ on the denominator of the fraction.

This isn't true: in general $(a-b)(a+b)=a^2-b^2$, not $a^2+b^2$.

Answer 2

$$(\sqrt6-2)(\sqrt6+2)=6-4=2\neq10$$

Answer 3

Recall when we have a product that expands to a "difference of squares", we obtain just that: a difference that we obtain, not a sum:

$$(a - b)(a+b) = a^2 - b^2$$ So your denominator should expand to $$(\sqrt 6 - 2)(\sqrt 6 + 2) = (\sqrt 6)^2 - 2^2 = 6 - 4 = 2.$$

Correcting for this gives us the correct answer: $$\dfrac{10 + 4\sqrt 6}{2} = 5 + 2\sqrt 6.$$