How do I know if a square root is irrational?

1 Answer
Jul 24, 2018

Answer:

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!