An irrational number is a real number or set of real numbers that cannot be written as a fraction of two integers (whole numbers). It is a non-terminating decimal that cannot be expressed as a fraction.
For example, $\sqrt{3}$ $\approx$ $1.73205…$ is a non-terminating decimal number which is irrational because it cannot be expressed as a fraction in the form $\frac a b$ where $a$ and $b$ are integers.
If the radical in the denominator is a square root, then you multiply by a square root that will give you a perfect square under the radical when multiplied by the denominator. If the radical in the denominator is a cube root, then you multiply by a cube root that will give you a perfect cube under the radical when multiplied by the denominator and so forth. Note that the phrase "perfect square" means that you can take the square root of it. Just as "perfect cube" means we can take the cube root of the number, and so forth.
Keep in mind that as long as you multiply the numerator and denominator by the exact same thing, the fractions will be equivalent.
Some radicals will already be in a simplified form, but make sure you simplify the ones that are not. If you need a review on this, go to Tutorial 39: Simplifying Radical Expressions.
Be careful. You cannot cancel out a factor that is on the outside of a radical with one that is on the inside of the radical. In order to cancel out common factors, they have to be both inside the same radical or be both outside the radical.
<aside> 1️⃣
$\frac{3}{1 + \sqrt{7}}$
</aside>
<aside> 2️⃣
$\frac{4\sqrt7+7\sqrt2}{\sqrt{14}}$
</aside>
<aside> 3️⃣
$\frac{2\sqrt{2}}{4-\sqrt6}$
</aside>
<aside> 4️⃣
$\frac{10}{3-2\sqrt{5}}$
</aside>
<aside> 5️⃣
$\frac{5}{7-4\sqrt{3}}$
</aside>
<aside> 6️⃣
$\frac{\sqrt{2}}{\sqrt6 +\sqrt3}$
</aside>
<aside> 7️⃣
$\frac{-6}{9+\sqrt{13}}$
</aside>
<aside> 8️⃣
$\frac{10-\sqrt5}{\sqrt5}$
</aside>