How do I know if a square root is irrational?

1 Answer
Jul 24, 2018

See below:

Explanation:

Let's say we have

n

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

54

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

96

36

We know that 6 is the same as 32, 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. 6 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:

4972214222355229814

We have expressed 49 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!