# Homework Help: Simplifying rational numbers help

1. Dec 22, 2009

### nobahar

Express
$$\frac{1+\sqrt{2}}{3-\sqrt{2}}$$ as $$a+b\sqrt{2}$$ where a and b are rational numbers.

I started by
$$\frac{1+\sqrt{2}}{3-\sqrt{2}} * \frac{3+\sqrt{2}}{3+\sqrt{2}}$$

But, I obtain

$$\frac{5}{7}-\frac{4}{7}\sqrt{2}$$

I believe that, here, a and b are rational, but is there a more tidy version? I tried playing with the square root so that it is a multiple of 2, so that I could 'split' it into two square roots, on of the square root of two so that I could use it to cancel out the square root of two on the bottom, then I could also remove the other square root if it was a rational square root. For example, I tried:

$$\frac{1+\sqrt{2}}{3-\sqrt{2}} * \frac{x+\sqrt{8}}{x+\sqrt{8}}$$
because
$$\sqrt{8}= \sqrt{2*4} = \sqrt{2} * \sqrt{4} = 2\sqrt{2}$$

I haven't specified x, since its just an example. I actually tried making a relationship between x and the square root I introduced, since I could represent it algebraically as
$$\sqrt{g}$$
where g is a multiple of two, would give
$$\sqrt{2}*\sqrt{\frac{1}{2}g}$$.

I'm guessing its more straight forward than this.

Last edited: Dec 22, 2009
2. Dec 22, 2009

### Staff: Mentor

Re: Simplifying

I don't believe there is. Assuming you make the correction I mentioned, you will have written the original expression in the form a + b*sqrt(2), where a and be are rational. You can't get any tidier than that.

3. Dec 22, 2009

### Mentallic

Re: Simplifying

Quite ambitious, but no. Since the original expression is irrational, if you're going to simplify it, it will always still be irrational. That is why the $\sqrt{2}$ must be there. You can of course "simplify" it into many different ways, but there wil always be an irrational portion of the expression.

4. Dec 23, 2009

### nobahar

Re: Simplifying

Many thanks Mark44 and Mentallic.