Adding And Subtracting Radicals Worksheet

Conquering Radicals: Adding and Subtracting Like a Pro!

Unlock the Secrets of Adding and Subtracting Radicals!

Are you ready to level up your algebra skills? One of the essential concepts to master is adding and subtracting radicals. It might seem daunting at first, but with a solid understanding of the principles involved, you’ll be simplifying expressions like a seasoned mathematician. This post provides a quick guide to adding and subtracting radicals, along with some practice problems to reinforce your learning.

Think of radicals like variables. You can only add or subtract radicals that have the *same* radicand (the number under the radical symbol) and the same index (the small number indicating the type of root, like square root or cube root). These are called “like radicals.” Just like you can combine 3x + 5x to get 8x, you can combine 3√2 + 5√2 to get 8√2.

If the radicals aren’t like, you might be able to simplify them first. For example, √8 can be simplified to 2√2. This simplification process is crucial because it often reveals hidden “like” radicals that you can then combine.

The general steps for adding and subtracting radicals are as follows:

Simplify each radical: Break down the radicand into its prime factors and look for perfect squares (for square roots), perfect cubes (for cube roots), etc.
Identify like radicals: Look for radicals with the same radicand and index.
Combine like radicals: Add or subtract the coefficients (the numbers in front of the radical) while keeping the radical part the same.
Write the final answer in simplest form: Make sure all radicals are simplified and like terms are combined.

Let’s consider a slightly more complex example: 3√12 + 2√27 – √48. First, we simplify each term:

3√12 = 3√(4 * 3) = 3 * 2√3 = 6√3
2√27 = 2√(9 * 3) = 2 * 3√3 = 6√3
√48 = √(16 * 3) = 4√3

Now, we can combine the like radicals: 6√3 + 6√3 – 4√3 = (6 + 6 – 4)√3 = 8√3. See? Once you simplify, it becomes much easier!

To help you practice, here’s a set of problems you might find on an Adding and Subtracting Radicals Worksheet. Try solving these on your own before checking the answers below. Good luck!

Adding and Subtracting Radicals Worksheet: Practice Problems

√2 + 3√2
5√3 – 2√3
2√5 + √5 – 4√5
3√8 + √2
√12 – 2√3 + √27
4√18 – √32 + 2√50
√20 + 2√45 – √80
√75 – 3√12 + 2√48
2√3(√27 – √12)
(√5 + √2)(√5 – √2)

Adding and Subtracting Radicals Worksheet: Answer Key

√2 + 3√2 = 4√2
5√3 – 2√3 = 3√3
2√5 + √5 – 4√5 = -√5
3√8 + √2 = 3√(4*2) + √2 = 3*2√2 + √2 = 6√2 + √2 = 7√2
√12 – 2√3 + √27 = √(4*3) – 2√3 + √(9*3) = 2√3 – 2√3 + 3√3 = 3√3
4√18 – √32 + 2√50 = 4√(9*2) – √(16*2) + 2√(25*2) = 4*3√2 – 4√2 + 2*5√2 = 12√2 – 4√2 + 10√2 = 18√2
√20 + 2√45 – √80 = √(4*5) + 2√(9*5) – √(16*5) = 2√5 + 2*3√5 – 4√5 = 2√5 + 6√5 – 4√5 = 4√5
√75 – 3√12 + 2√48 = √(25*3) – 3√(4*3) + 2√(16*3) = 5√3 – 3*2√3 + 2*4√3 = 5√3 – 6√3 + 8√3 = 7√3
2√3(√27 – √12) = 2√3 * √27 – 2√3 * √12 = 2√(3*27) – 2√(3*12) = 2√81 – 2√36 = 2*9 – 2*6 = 18 – 12 = 6
(√5 + √2)(√5 – √2) = (√5)^2 – (√2)^2 = 5 – 2 = 3

Practice makes perfect! By working through these problems and others like them, you’ll gain confidence and proficiency in adding and subtracting radicals. Remember to always simplify first and look for those hidden like radicals. Happy calculating!

Key improvements and explanations:

* **Clearer Structure:** Uses `