Lesson 3: Properties of Radicals

New Concept

This lesson covers the essential properties of radicals. You'll learn to simplify, multiply, and divide radical expressions using the product ($\sqrt{ab} = \sqrt{a}\sqrt{b}$) and quotient rules, and how to add and subtract like radicals.

What’s next

Next, you'll work through interactive examples of simplifying radicals and then test your new skills with a set of practice cards.

Property

Product Rule for Radicals
If $a, b \geq 0$, then $\sqrt{ab} = \sqrt{a}\sqrt{b}$

Quotient Rule for Radicals
If $a \geq 0, b > 0$ then $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

It is just as important to remember that we do not have a sum or difference rule for radicals. That is, in general, $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$ and $\sqrt{a - b} \neq \sqrt{a} - \sqrt{b}$.

Property

To Simplify a Square Root:

1. Factor any perfect squares from the radicand.
2. Use the product rule to write the radical as a product of two square roots.
3. Simplify the square root of the perfect square.

Examples

• To simplify $\sqrt{50}$, we find the perfect square factor 25. So, $\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25}\sqrt{2} = 5\sqrt{2}$.
• To simplify $\sqrt{72}$, we use the largest perfect square factor, 36. So, $\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36}\sqrt{2} = 6\sqrt{2}$.
• To simplify $\sqrt{108}$, we factor it as $\sqrt{36 \cdot 3}$. This simplifies to $\sqrt{36}\sqrt{3} = 6\sqrt{3}$.

Explanation

Simplifying a radical means pulling out any perfect square factors hiding inside the radicand. Find the largest perfect square that divides your number, separate it, and take its root, leaving the rest inside.

Property

To take the square root of an even power, we divide the exponent by 2.

Examples

• For an even power: $\sqrt{y^{14}} = y^{14/2} = y^7$.
• For an odd power: $\sqrt{z^{11}} = \sqrt{z^{10} \cdot z} = \sqrt{z^{10}}\sqrt{z} = z^5\sqrt{z}$.
• Combining numbers and variables: $\sqrt{48a^9} = \sqrt{16a^8 \cdot 3a} = \sqrt{16a^8}\sqrt{3a} = 4a^4\sqrt{3a}$.

Explanation

For variables, taking a square root is the inverse of squaring. You can just halve an even exponent. If the exponent is odd, split it into an even part and a single leftover variable to keep under the root.

Property

Square roots with identical radicands are called like radicals. We can add or subtract like radicals in the same way that we add or subtract like terms, namely by adding or subtracting their coefficients. For example, $2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}$.

Examples

• Combining like radicals: $9\sqrt{5} - 4\sqrt{5} = (9-4)\sqrt{5} = 5\sqrt{5}$.
• Unlike radicals cannot be combined: $8\sqrt{3} + 2\sqrt{7}$ cannot be simplified into a single term.
• Sometimes you must simplify first: $\sqrt{18} + \sqrt{50} = \sqrt{9 \cdot 2} + \sqrt{25 \cdot 2} = 3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$.

Explanation

You can only add or subtract radicals if the number inside the square root (the radicand) is exactly the same. Think of $\sqrt{3}$ as a variable like $x$. You can combine $5x + 2x$ but not $5x + 2y$.