How do I know if a square root is irrational?

Jul 24, 2018

See below:

Explanation:

Let's say we have

$\sqrt{n}$

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

$\sqrt{54}$

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

$\sqrt{9} \cdot \sqrt{6}$

$\implies 3 \sqrt{6}$

We know that $6$ is the same as $3 \cdot 2$, 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. $\sqrt{6}$ 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:

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

We have expressed $\sqrt{49}$ 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!