combine and simplify these radicals. 160 40

2 min read 28-02-2025

combine and simplify these radicals. 160 40

Combining and Simplifying Radicals: A Step-by-Step Guide (160 and 40)

This article will guide you through the process of combining and simplifying the radicals √160 and √40. We'll break down each step, showing you how to find the simplest radical form. Understanding this process is crucial for working with radicals in algebra and beyond.

Understanding Radicals

Before we begin, let's refresh our understanding of radicals. A radical expression contains a radical symbol (√), indicating a root (usually a square root). The number inside the radical symbol is called the radicand. Simplifying radicals means expressing them in their most concise form, without any perfect square factors remaining under the radical.

Step 1: Prime Factorization of the Radicands

The first step in simplifying radicals is to find the prime factorization of each radicand (the number inside the square root).

√160: The prime factorization of 160 is 2 x 2 x 2 x 2 x 2 x 5 = 2⁵ x 5
√40: The prime factorization of 40 is 2 x 2 x 2 x 5 = 2³ x 5

Step 2: Identifying Perfect Squares

Now, we look for perfect squares within the prime factorizations. A perfect square is a number that is the result of squaring an integer (e.g., 4 = 2², 9 = 3², 16 = 4²).

√160: We can rewrite the prime factorization as √(2⁴ x 2 x 5) Notice 2⁴ is a perfect square (2² x 2² = 16).
√40: We can rewrite the prime factorization as √(2² x 2 x 5). Here, 2² is a perfect square (4).

Step 3: Simplifying the Radicals

Now, we simplify each radical using the property √(a x b) = √a x √b:

√160: √(2⁴ x 2 x 5) = √2⁴ x √(2 x 5) = 2²√10 = 4√10
√40: √(2² x 2 x 5) = √2² x √(2 x 5) = 2√10

Step 4: Combining Like Radicals

Both simplified radicals now contain √10. Since they have the same radicand, we can combine them:

4√10 + 2√10 = 6√10

Final Answer: Simplifying √160 + √40

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

Practice Problems

Try simplifying these expressions on your own:

√72 + √98
√27 + √12
√200 - √8

By following these steps, you'll be able to confidently combine and simplify radicals in your math problems. Remember, the key is prime factorization and identifying perfect squares within the radicands. Practice makes perfect!