Simplifying Radicals & Rationalizing Denominators Cheat Sheet

Key Facts

• A radical is simplified when the radicand has no perfect-square factor greater than $1$, such as $\sqrt{72}=6\sqrt{2}$.
• The product rule for square roots is $\sqrt{a}\sqrt{b}=\sqrt{ab}$ when $a\ge 0$ and $b\ge 0$.
• The quotient rule for square roots is $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$ when $a\ge 0$ and $b>0$.
• To simplify $\sqrt{a^2}$, use $\sqrt{a^2}=|a|$ because a square root represents the principal nonnegative root.
• Like radicals have the same radical part, so $3\sqrt{5}+2\sqrt{5}=5\sqrt{5}$ but $3\sqrt{5}+2\sqrt{3}$ cannot be combined.
• To rationalize $\frac{a}{\sqrt{b}}$, multiply by $\frac{\sqrt{b}}{\sqrt{b}}$ to get $\frac{a\sqrt{b}}{b}$.
• To rationalize $\frac{a}{b+\sqrt{c}}$, multiply by the conjugate $\frac{b-\sqrt{c}}{b-\sqrt{c}}$.
• Conjugates remove radicals from binomial denominators because $(b+\sqrt{c})(b-\sqrt{c})=b^2-c$.

Vocabulary

Radical

A radical is an expression with a root symbol, such as $\sqrt{18}$.

Radicand

The radicand is the number or expression inside the radical symbol, such as $18$ in $\sqrt{18}$.

Perfect square

A perfect square is a number that can be written as $n^2$ for an integer $n$, such as $25=5^2$.

Simplest radical form

A radical is in simplest radical form when no perfect-square factor remains inside the radical and no radical remains in the denominator.

Rationalizing the denominator

Rationalizing the denominator means rewriting a fraction so that the denominator contains no radical.

Conjugate

The conjugate of $a+\sqrt{b}$ is $a-\sqrt{b}$, and their product is $a^2-b$.