How do I know if a square root is irrational?

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How do I know if a square root is irrational?

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Answer 1 · 2018-07-24T03:14:58

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!

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How do I know if a square root is irrational?

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