combine and simplify these radicals. 160 40

2 min read 28-02-2025

combine and simplify these radicals. 160 40

Understanding how to combine and simplify radicals is a fundamental skill in algebra. This article will guide you through the process, using the example of simplifying the radicals √160 and √40. We'll break down the steps clearly, so you can confidently tackle similar problems.

What are Radicals?

Before we dive into combining and simplifying, let's quickly define what a radical is. A radical expression contains a radical symbol (√), indicating a root (like a square root, cube root, etc.) of a number. The number under the radical symbol is called the radicand.

Step 1: Finding the Prime Factorization

The key to simplifying radicals lies in finding the prime factorization of the radicand. Prime factorization means expressing a number as a product of its prime factors (numbers only divisible by 1 and themselves).

Let's find the prime factorization of 160:

160 = 2 x 80 = 2 x 2 x 40 = 2 x 2 x 2 x 20 = 2 x 2 x 2 x 2 x 10 = 2 x 2 x 2 x 2 x 2 x 5 = 2⁵ x 5

Now let's do the same for 40:

40 = 2 x 20 = 2 x 2 x 10 = 2 x 2 x 2 x 5 = 2³ x 5

Step 2: Simplifying the Radicals

Once you have the prime factorization, you can simplify the radicals. Look for pairs (or triples for cube roots, quadruples for fourth roots, etc.) of the same prime factor. Each pair (or group) can be brought outside the radical as a single factor.

For √160:

√160 = √(2⁵ x 5) = √(2⁴ x 2 x 5) = √2⁴ x √(2 x 5) = 2²√10 = 4√10

For √40:

√40 = √(2³ x 5) = √(2² x 2 x 5) = √2² x √(2 x 5) = 2√10

Step 3: Combining Like Radicals

Now that we've simplified both radicals, notice that they both have the same radicand, √10. This makes them "like radicals," meaning they can be combined just like like terms in algebra.

√160 + √40 = 4√10 + 2√10 = (4 + 2)√10 = 6√10

Therefore, the simplified and combined form of √160 + √40 is 6√10.

Practice Problems

Try simplifying these on your own:

√72 + √50
√27 - √12
√128 + √32

Remember to follow the steps: prime factorization, simplification, and combination of like radicals.

Conclusion

Combining and simplifying radicals involves a systematic approach. By breaking down the radicands into their prime factors and then extracting perfect squares (or cubes, etc.), you can significantly simplify complex radical expressions. Remember the key is to look for perfect squares (or cubes, etc.) within the radicand to simplify. Practice makes perfect, so work through several examples to build your confidence and understanding. You now have the skills to effectively simplify and combine radicals, making your algebraic calculations cleaner and more efficient.