Simplifying Radicals

What is a Radical?

A radical (square root) asks: “what number, multiplied by itself, gives this?”

$\sqrt{9} = 3 \quad \text{because} \quad 3 \times 3 = 9$

$\sqrt{25} = 5 \quad \text{because} \quad 5 \times 5 = 25$

$\sqrt{ab} = \sqrt{a} \times \sqrt{b}$

You can split a radical into a product of radicals.

This lets us pull out perfect squares.

Example: Simplify $\sqrt{12}$

12 isn’t a perfect square. But $12 = 4 \times 3$ , and 4 is.

\begin{aligned} \sqrt{12} &= \sqrt{4 \times 3} \\ &= \sqrt{4} \times \sqrt{3} \\ &= 2\sqrt{3} \end{aligned}

Example: Simplify $\sqrt{50}$

$50 = 25 \times 2$

\begin{aligned} \sqrt{50} &= \sqrt{25 \times 2} \\ &= 5\sqrt{2} \end{aligned}

Example: Simplify $\sqrt{72}$

$72 = 36 \times 2$

\begin{aligned} \sqrt{72} &= \sqrt{36 \times 2} \\ &= 6\sqrt{2} \end{aligned}

You can simplify in steps. Same answer, just longer.

\begin{aligned} \sqrt{72} &= \sqrt{4 \times 18} \\ &= 2\sqrt{18} \\ &= 2\sqrt{9 \times 2} \\ &= 2 \times 3\sqrt{2} \\ &= 6\sqrt{2} \end{aligned}

Same rules. Pull out pairs.

$\sqrt{x^2} = x$

$\sqrt{x^4} = x^2$

$\sqrt{x^3} = \sqrt{x^2 \times x} = x\sqrt{x}$

Example: Simplify $\sqrt{18x^3}$

\begin{aligned} \sqrt{18x^3} &= \sqrt{9 \times 2 \times x^2 \times x} \\ &= 3x\sqrt{2x} \end{aligned}

Only combine like radicals (same number under the root).

$2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3}$

$2\sqrt{3} + 5\sqrt{2} = \text{can't simplify}$

Sometimes you need to simplify first:

\begin{aligned} \sqrt{12} + \sqrt{27} &= 2\sqrt{3} + 3\sqrt{3} \\ &= 5\sqrt{3} \end{aligned}

No radicals in the denominator.

Multiply top and bottom by the radical to clear it.

Example: Rationalize $\dfrac{1}{\sqrt{2}}$

\begin{aligned} \frac{1}{\sqrt{2}} &= \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} \\[0.5em] &= \frac{\sqrt{2}}{2} \end{aligned}

Example: Rationalize $\dfrac{3}{\sqrt{5}}$

\begin{aligned} \frac{3}{\sqrt{5}} &= \frac{3}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} \\[0.5em] &= \frac{3\sqrt{5}}{5} \end{aligned}