Question

How do I know if a square root is irrational?

Answer 1

See below:

Explanation:

Let's say we have

`sqrtn`

This is irrational if `n` is a prime number, or has no perfect square factors.

When we simplify radicals, we try to factor out perfect squares. For example, if we had

`sqrt(54)`

We know that `9` is a perfect square, so we can rewrite this as

`sqrt9*sqrt6`

`=>3sqrt6`

We know that `6` is the same as `3*2`, but neither of those numbers are perfect squares, so we can't simplify this further.

Remember what irrational means- we cannot express the number as a ratio of two other numbers. `sqrt6` just continues on and on forever, which is what makes it irrational.

We know that the principal root of `49` is `7`. What makes this rational is that we can express `7` in many ways:

#49/7color(white)(2/2)14/2color(white)(2/2)35/5color(white)(2/2) 98/14#

We have expressed `sqrt49` as the ratio of two integers. This is what makes a number rational.

For a much better explanation on why the square root of a prime number is irrational, I strongly encourage you to check out this Khan Academy video.

Hope this helps!