Lesson 69: Adding and Subtracting Radical Expressions

New Concept

Like radicals have the same radicand and index. Unlike radicals have different radicands and/or index numbers.

What’s next

Next, you’ll apply this concept to combine and simplify radical expressions, even when they don’t look alike at first.

Property

Like radicals have the same radicand and index. Unlike radicals have different radicands and/or index numbers.

Explanation

Just like you can’t combine apples and bananas, you can't combine unlike radicals. For radicals to be 'like,' they must have the exact same number under the root sign (radicand) and the same root type (index). They must be perfect matches to be grouped together for addition or subtraction. It’s all about finding those radical twins!

Examples

• $5\sqrt{3}$ and $2\sqrt{3}$ are like radicals because they both have the radicand $3$.
• $6\sqrt{2}$ and $8\sqrt{11}$ are unlike radicals because the radicands $2$ and $11$ are different.
• $4\sqrt{xy}$ and $-6\sqrt{xy}$ are like radicals because the radicand $xy$ is the same in both.

Property

To combine like radicals, add or subtract their coefficients, just like with like terms: $a\sqrt{x} + b\sqrt{x} = (a+b)\sqrt{x}$.

Explanation

Once you’ve found your 'radical twins' (like radicals), combining them is simple! Just add or subtract the numbers in front, called coefficients, and keep the radical part the same. This works exactly like combining variables, such as $2x + 4x = 6x$. The radical is treated just like the variable part of a term.

Examples

• $2\sqrt{7} + 4\sqrt{7} = (2+4)\sqrt{7} = 6\sqrt{7}$
• $4\sqrt{xy} - 6\sqrt{xy} = (4-6)\sqrt{xy} = -2\sqrt{xy}$
• $10\sqrt{3b} - 2\sqrt{3b} = (10-2)\sqrt{3b} = 8\sqrt{3b}$

Adding radicals is just like combining like terms—if the core part matches, they mix. This first key idea, combining like radicals, is the foundation for everything else in this lesson.

Example Problem
Simplify the expression $\frac{5\sqrt{3p}}{9} + \frac{2\sqrt{3p}}{9} - \frac{6\sqrt{2q}}{9}$.

Step-by-Step

1. First, we identify the like radicals in the expression. The terms with $\sqrt{3p}$ are like radicals.

2. We can combine the coefficients of the like radicals since they share a common denominator.

3. The term with $\sqrt{2q}$ has a different radicand, so it is an unlike radical and cannot be combined further.

Property

All radicals should be simplified before trying to identify like radicals. Use the Product Property of Radicals: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$.

Explanation

Sometimes radicals are like secret twins in disguise! You must simplify them first by pulling out any perfect square factors from the radicand. This process often reveals that radicals you thought were different are actually alike. Always simplify first, then hunt for like radicals to combine. Don't judge a radical by its cover until you've simplified it!

Examples

• $\sqrt{12} + \sqrt{75} = \sqrt{4 \cdot 3} + \sqrt{25 \cdot 3} = 2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3}$
• $5\sqrt{18} - 2\sqrt{2} = 5\sqrt{9 \cdot 2} - 2\sqrt{2} = 5 \cdot 3\sqrt{2} - 2\sqrt{2} = 15\sqrt{2} - 2\sqrt{2} = 13\sqrt{2}$
• $c\sqrt{75c} - \sqrt{27c^3} = c\sqrt{25 \cdot 3c} - \sqrt{9c^2 \cdot 3c} = 5c\sqrt{3c} - 3c\sqrt{3c} = 2c\sqrt{3c}$

Sometimes radicals are in disguise. Let's unmask them to see if they're actually alike. The second key idea is to simplify first, which is often necessary before you can combine terms.

Example Problem
Simplify $5\sqrt{12k^3} + 3\sqrt{3k} + 7\sqrt{3k}$, where $k$ is a non-negative real number.

Step-by-Step

1. We examine the expression and see that the first radical, $\sqrt{12k^3}$, might be simplified. The other two, $\sqrt{3k}$, are already in simplest form.

2. Factor the first radicand, $12k^3$, looking for perfect squares.

3. Apply the Product Property of Radicals to separate the factors.

4. Simplify the perfect square roots. Remember that $5 \cdot \sqrt{4} \cdot \sqrt{k^2}$ becomes $5 \cdot 2 \cdot k$.

5. Now all terms are like radicals. Factor out the common radical, $\sqrt{3k}$.

6. Simplify the expression inside the parentheses.