Let $a = \frac{9+\sqrt{45}}{2}$. Find $\frac{1}{a}$

Question

I've been wrapping my head around this question lately:

Let

$$a = \frac{9+\sqrt{45}}{2}$$

Find the value of

$$\frac{1}{a}$$

I've done it like this:

$$\frac{1}{a}=\frac{2}{9+\sqrt{45}}$$

I rationalize the denominator like this:

$$\frac{1}{a}=\frac{2}{9+\sqrt{45}} \times (\frac{9-\sqrt{45}}{9-\sqrt{45}})$$

This is what I should get:

$$\frac{1}{a} = \frac{2(9-\sqrt{45})}{81-45} \rightarrow \frac{1}{a}=\frac{18-2\sqrt{45}}{36})$$

Which I can simplify to:

$$\frac{1}{a}=\frac{2(9-\sqrt{45})}{36}\rightarrow\frac{1}{a}=\frac{9-\sqrt{45}}{18}$$

However, this answer can't be found in my multiple choice question here:

Any hints on what I'm doing wrong?

Answer 1

$$\frac { 1 }{ a } =\frac { 9-\sqrt { 45 } }{ 18 } =\frac { 3\left( 3-\sqrt { 5 } \right) }{ 18 } =\frac { 3-\sqrt { 5 } }{ 6 } $$

Answer 2

The answer is b, which is equivalent to your answer after simplifying.

This is because $ 9 - \sqrt{45} = 9 - \sqrt{9 * 5} = 9 - 3\sqrt{5}$. Then,$ \frac{9 - 3\sqrt{5}}{18} = \frac{3 - \sqrt{5}}{6}. $

Answer 3

Since it is a multiple choice question, you can just multiply the given term with all the possible terms and verify when you get $1$