Skill: Simplify radicals containing perfect-square factors

A radical is in simplest form when its radicand has no perfect-square factor greater than $1$.

To simplify a numerical radical:

1. Find a perfect-square factor of the radicand.
2. Rewrite the radicand as a product.
3. Use $\sqrt{ab}=\sqrt a\sqrt b$.
4. Take the square root of the perfect square.

Example: Simplify $\sqrt{72}$.

First, factor $(72)$ using the largest perfect square factor:

$$72=36\cdot 2$$

Rewrite the radical:

$$\sqrt{72}=\sqrt{36\cdot 2}$$

Separate the product under the radical:

$$\sqrt{36\cdot 2}=\sqrt{36}\sqrt{2}$$

Since $\sqrt{36}=6$, simplify:

$$\boxed{\sqrt{72}=6\sqrt{2}}$$

The answer is in simplest form because $2$ has no perfect-square factor greater than $1$.

Remember, this property applies to multiplication inside a radical, not addition. For example, $\sqrt{36+2}$ cannot be changed to $\sqrt{36}+\sqrt{2}$.