Lesson 61: Simplifying Radical Expressions

New Concept

If $a$ and $b$ are non-negative real numbers, then:

What’s next

Next, you’ll apply this rule to simplify radicals using perfect squares, prime factors, and variables to solve problems.

Property

If $a$ and $b$ are non-negative real numbers, then

Explanation

Think of radicals as party invitations! You can either send one big invitation ($\sqrt{ab}$) or separate ones for each guest ($\sqrt{a} \cdot \sqrt{b}$). This rule lets you break down big, scary-looking square roots into smaller, friendlier pieces, making them much easier to solve. It's all about teamwork for simplification!

Examples

$\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}$
$\sqrt{3} \cdot \sqrt{12} = \sqrt{3 \cdot 12} = \sqrt{36} = 6$
$\sqrt{2} \cdot \sqrt{18} = \sqrt{2 \cdot 18} = \sqrt{36} = 6$

Property

A perfect square is a number that is the square of an integer. To simplify, factor the radicand to find its largest perfect square factor. Use the property $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$.

Explanation

Go on a treasure hunt inside the radical! Your goal is to find 'perfect squares'—numbers like 4, 9, and 25. When you find one, you can pull its square root out of the radical sign. Any numbers left behind that aren't perfect squares have to stay inside. It's the great radical escape!

Examples

$\sqrt{75} = \sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3}$
$\sqrt{200} = \sqrt{100 \cdot 2} = \sqrt{100} \cdot \sqrt{2} = 10\sqrt{2}$
$\sqrt{147} = \sqrt{49 \cdot 3} = \sqrt{49} \cdot \sqrt{3} = 7\sqrt{3}$

Let's break down this radical to its simplest form using its hidden perfect squares. This example demonstrates the key idea of simplifying with perfect squares.

Example Problem: Simplify $\sqrt{108}$.

1. We start by looking for perfect square factors of $108$. We can see that $108$ is divisible by $9$ and $4$. So we can write $108 = 9 \cdot 4 \cdot 3$.
2. Rewrite the radical using these factors:

3. Now, apply the Product Property of Radicals to separate the square roots:

4. Simplify the roots of the perfect squares:

5. Finally, multiply the whole numbers outside the radical: