How do you simplify `1/sqrt7`?

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.

`1/sqrt(7) xx sqrt(7)/sqrt(7)`

`= sqrt(7)/sqrt(49)`

`= 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 xx sqrt(7) = 3sqrt(7)` while `sqrt(3) xx sqrt(7) = sqrt(21)`

a) `4/sqrt(5)`

b) `(3 + sqrt(6))/sqrt(2)`

c) `(5 - sqrt(7))/(sqrt(10) - sqrt(11))`

d) `(sqrt(3) + sqrt(2))/(5 + sqrt(6))`

2. Challenge question:

Rationalise the denominator of `sqrt(2)/sqrt((x^2 + 6x + 5))`