- #1

nobahar

- 497

- 2

Express

[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.

[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.

Last edited: