Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Mini Lesson Question

Question #1: Simplifying Radicals

Simplify

\sqrt{45}

.

$5\sqrt{9}$
$9\sqrt{5}$
$5\sqrt{3}$
$3\sqrt{5}$

Simplifying a Square Root

Some square roots can be simplified using the product property. Here is the product property:

Product Property of $n^{t h}$ Roots

If $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers, and $n \geq 2$ is an integer, then

$\sqrt[n]{a b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ and $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a b}$

Here is how the product property can be used to simplify square roots:

How to Simplify a Radical Expression Using the Product Property

Step 1 - Find the largest factor in the radicand that is a perfect power of the index. Use that factor to rewrite the radicand as a product of two factors.

Step 2 - Use the product rule to rewrite the radical as the product of two radicals.

Step 3 - Simplify the root of the perfect power.

Example

Find the simplest form of $\sqrt{98}$ .

Step 1 - Find the largest factor in the radicand that is a perfect power of the index. Use that factor to rewrite the radicand as a product of two factors. We see that 49 is the largest factor of 98 that has a power of 2. In other words, 49 is the largest perfect square factor of 98.

$98 = 49 \cdot 2$ Always write the perfect square factor first. $\sqrt{49 \cdot 2}$

Step 2 - Use the product rule to rewrite the radical as the product of two radicals.

$\sqrt{49} \cdot \sqrt{2}$

Step 3 - Simplify the root of the perfect power.

$7 \sqrt{2}$

Try it

Try It: Simplifying a Square Root

Simplify $\sqrt{48}$ .

Here is how to simplify a square root:

Step 1 - Find the largest factor in the radicand and rewrite the radical as a product of factors.

$\sqrt{48} = \sqrt{16 \cdot 3}$

Step 2 - Use the product rule to rewrite the radical as a product of two radicals.

$\sqrt{16} \cdot \sqrt{3}$

Step 3 - Simplify the radical with the perfect square root.

$\sqrt{16} \cdot \sqrt{3} = 4 \sqrt{3}$

Check Your Understanding

Find the simplest form of $\sqrt{500}$ .

Multiple Choice:

$10 \sqrt{5}$
$100 \sqrt{5}$
$5 \sqrt{20}$
$2 \sqrt{125}$

Check yourself: The $\sqrt{500}$ can be broken into $\sqrt{100} \cdot \sqrt{5}$ . This simplifies to $10 \sqrt{5}$ .

Video: Simplifying Square Roots Using Perfect Squares

Watch the following video to learn more about how to simplify square roots using perfect squares.

Khan Academy: Simplifying Square Roots

Access multimedia content

Simplifying Radicals: Mini-Lesson Review