combine these radicals.

Lyra Rowan

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29-09-2024

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Combining Radicals: A Step-by-Step Guide

Combining radicals can seem daunting at first, but with the right approach, it becomes a simple process. Understanding the rules and applying them systematically will empower you to simplify radical expressions and arrive at the most efficient solutions. This article will explore the key concepts of combining radicals, using examples and explanations drawn from BrainlY, a collaborative learning platform.

What are radicals?

Radicals, also known as roots, are the opposite of exponents. They represent the value that, when multiplied by itself a certain number of times, results in the original number. For example, the square root of 9 is 3, because 3 x 3 = 9.

Key Rules for Combining Radicals

• Only like radicals can be combined. Like radicals have the same index (the small number outside the radical symbol) and the same radicand (the number under the radical symbol). For instance, √2 and 3√2 are like radicals, but √2 and √3 are not.

• Combine coefficients. When combining like radicals, add or subtract their coefficients (the number in front of the radical). For example, 2√5 + 3√5 = (2+3)√5 = 5√5.

Example 1: Combining Square Roots

Question: Simplify the expression: 2√12 + 5√3 - √48

Answer:

First, simplify the radicals to their simplest form:

• √12 = √(4 x 3) = 2√3
• √48 = √(16 x 3) = 4√3

Now we can rewrite the expression:

2√12 + 5√3 - √48 = 2(2√3) + 5√3 - 4√3 = 4√3 + 5√3 - 4√3

Combine the coefficients:

4√3 + 5√3 - 4√3 = (4 + 5 - 4)√3 = 5√3

Therefore, the simplified expression is 5√3.

Example 2: Combining Cube Roots

Question: Simplify: ∛27 + 2∛8 - ∛64

Answer:

• ∛27 = 3 (because 3 x 3 x 3 = 27)
• ∛8 = 2 (because 2 x 2 x 2 = 8)
• ∛64 = 4 (because 4 x 4 x 4 = 64)

The expression becomes:

∛27 + 2∛8 - ∛64 = 3 + 2(2) - 4 = 3 + 4 - 4 = 3

Therefore, the simplified expression is 3.

Additional Considerations

• Factoring: Sometimes, you need to factor the radicand to simplify it before combining. For example, √27 = √(9 x 3) = 3√3.

• Variables: The same rules apply to radicals with variables. For example, 2√x + 5√x = 7√x.

Practicing and Mastering Combining Radicals

Combining radicals is a crucial skill in algebra. It's important to practice applying the rules and simplifying expressions to build confidence and understanding. You can use BrainlY as a resource to find practice problems and ask for help when needed.

Remember, the key to combining radicals is to identify like radicals, simplify them to their simplest form, and then combine the coefficients.