# How do you simplify 1/sqrt7?

Feb 19, 2016

All you can do is rationalize the denominator.

#### Explanation:

To rationalize the denominator just means to make the denominator a rational number. We can do this by multiplying the numerator and the denominator by $\sqrt{7}$, in order to keep the expression equivalent.

$\frac{1}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}}$

$= \frac{\sqrt{7}}{\sqrt{49}}$

$= \frac{\sqrt{7}}{7}$

This is as far as we can simplify. Here are the two rules about rationalizing a denominator:

When a monomial (one term in the denominator): Multiply the numerator and the denominator by the radical in the original expression's denominator

When a binomial (two terms in the denominator): Multiply the numerator and the denominator by the conjugate of the radical in the original expression. The conjugate forms a difference of squares. Example: $\sqrt{2} + 4$ is the conjugate of $\sqrt{2} - 4$. Essentially, you must switch the sign in the middle.

Practice exercises:

1. Simplify completely. Don't forget: you can only multiply non radicals with non radicals and radicals with radicals. Example: $3 \times \sqrt{7} = 3 \sqrt{7}$ while $\sqrt{3} \times \sqrt{7} = \sqrt{21}$

a) $\frac{4}{\sqrt{5}}$

b) $\frac{3 + \sqrt{6}}{\sqrt{2}}$

c) $\frac{5 - \sqrt{7}}{\sqrt{10} - \sqrt{11}}$

d) $\frac{\sqrt{3} + \sqrt{2}}{5 + \sqrt{6}}$

2. Challenge question:

Rationalise the denominator of $\frac{\sqrt{2}}{\sqrt{\left({x}^{2} + 6 x + 5\right)}}$