Express(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\frac{1+\sqrt{2}}{3-\sqrt{2}}[/tex] as [tex]a+b\sqrt{2}[/tex] where a and b are rational numbers.

I started by

[tex]\frac{1+\sqrt{2}}{3-\sqrt{2}} * \frac{3+\sqrt{2}}{3+\sqrt{2}}[/tex]

But, I obtain

[tex]\frac{5}{7}-\frac{4}{7}\sqrt{2}[/tex]

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:

[tex]\frac{1+\sqrt{2}}{3-\sqrt{2}} * \frac{x+\sqrt{8}}{x+\sqrt{8}}[/tex]

because

[tex]\sqrt{8}= \sqrt{2*4} = \sqrt{2} * \sqrt{4} = 2\sqrt{2}[/tex]

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

[tex]\sqrt{g}[/tex]

where g is a multiple of two, would give

[tex]\sqrt{2}*\sqrt{\frac{1}{2}g}[/tex].

I'm guessing its more straight forward than this.

Thanks in advance.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Simplifying rational numbers help

**Physics Forums | Science Articles, Homework Help, Discussion**