# Simplifying rational numbers help

• nobahar
In summary, the expression \frac{1+\sqrt{2}}{3-\sqrt{2}} can be expressed as a+b\sqrt{2} where a and b are rational numbers. After manipulation, the final form is \frac{5}{7}+\frac{4}{7}\sqrt{2}. While there may be different ways to simplify this expression, the irrational portion represented by \sqrt{2} will always remain.
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:

nobahar said:
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

$$\frac{5}{7}-\frac{4}{7}\sqrt{2}$$
nobahar said:
I believe that, here, a and b are rational, but is there a more tidy version?
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.
nobahar said:
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.

nobahar said:
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.
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.

Many thanks Mark44 and Mentallic.

## What is a rational number?

A rational number is any number that can be expressed as a ratio of two integers, where the denominator is not equal to zero.

## Why is it important to simplify rational numbers?

Simplifying rational numbers makes them easier to work with and understand. It also helps to identify equivalent fractions and reduce the chance of errors in calculations.

## How do you simplify a rational number?

To simplify a rational number, divide both the numerator and denominator by their greatest common factor (GCF). Keep dividing until the GCF is 1 or you cannot divide any further.

## What is the difference between simplifying and reducing a rational number?

Simplifying a rational number is the process of dividing both the numerator and denominator by their GCF. Reducing a rational number is the process of finding an equivalent fraction with the smallest possible numbers.

## Can all rational numbers be simplified?

Yes, all rational numbers can be simplified. However, some may not have a simplified form because the numerator and denominator have no common factors other than 1.

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