Radicals & Roots

Simplify square roots and rationalize a denominator.

The idea

The square root $\sqrt{n}$ of a number $n \ge 0$ is the nonnegative number whose square is $n$. Nonnegative is part of the definition: both $3$ and $-3$ square to $9$, and the symbol is agreed to mean $3$, so $\sqrt{9}$ names one number.

When $n$ is a perfect square, the root is a whole number. Otherwise it is usually irrational — not equal to any ratio of two integers, as $\sqrt{2}$ is not — so no decimal we could write is exactly right, and we record the number by leaving the root standing and simplifying around it.

Simplifying rests on one rule: for $x, y \ge 0$,

$\sqrt{xy} = \sqrt{x}\,\sqrt{y}.$

To simplify a root, split off the largest perfect-square factor and take its root outside, leaving $a\sqrt{b}$ with no square factor inside. The same rule clears a root out of a denominator: multiply the numerator and the denominator by that root. We multiplied by $1$ in disguise, so the value is unchanged; only its form is.

Ways to work on it

• Walkthrough. Evaluate, simplify, and rationalize square roots.
• Practice. Simplify a square root into a√b form.
• Hardest. Combine like radicals or pull out a large square factor.

Not sure where to start? Take the ten-question placement test.