Make the denominator rational

Question

I am doing the homework of a recorded video in a course, and I came across this question which I am not able to solve:

Simplify the denominator: $\dfrac{12}{3+\sqrt5-2\sqrt2}$

And I am not able to understand the solution:

Thanks in advance.

Answer 1

What we are doing is a process called "rationalizing" the denominator. The process relies on the property that

$$( a + b)\cdot ( a - b) = a^2 - b^2 $$

So when you have a fraction as follows

$$F = \frac{12}{3+\sqrt5 - 2\sqrt2}$$

Let $a = 3+\sqrt5$, $b = 2\sqrt2$

$$F = \frac{12}{a-b}$$

Now multiplying numerator and denominator by $a+b$, we have

$$F = \frac{12(a+b)}{(a-b)(a+b)} = \frac{12(3+\sqrt5 + 2\sqrt2)}{(3+\sqrt5)^2-(2\sqrt2)^2} = \frac{12(3+\sqrt5+2\sqrt2)}{6(1+\sqrt5) }$$

$$\implies F =\frac{2(3+\sqrt5+2\sqrt2)}{1+\sqrt5} $$

Now you can repeat the process to simplify it further