https://file.aiquickdraw.com/mathbot/file/1726806397341eve6ah2g.jpg

The problem asks to simplify the radical expression \(\sqrt{224}\). Let's go step by step:

### Step 1: Prime Factorization
First, find the prime factorization of 224:
\[
224 \div 2 = 112 \\
112 \div 2 = 56 \\
56 \div 2 = 28 \\
28 \div 2 = 14 \\
14 \div 2 = 7 \\
7 \div 7 = 1
\]
So, the prime factorization of 224 is:
\[
224 = 2^5 \times 7
\]

### Step 2: Simplify the Radical
We can now take out any square factors from under the square root. Since \(2^4 = (2^2)^2\), we can take \(2^2 = 4\) outside the square root:
\[
\sqrt{224} = \sqrt{2^5 \times 7} = \sqrt{(2^4) \times 2 \times 7} = 2^2 \times \sqrt{2 \times 7} = 4\sqrt{14}
\]

So, \(\sqrt{224} = 4\sqrt{14}\).

You can input the simplified form \(4\sqrt{14}\) as your answer.

---

Would you like more details on this? Here are some follow-up questions to consider:

1. What is the significance of prime factorization in simplifying square roots?
2. Can any other radicals be simplified in the same manner as \(\sqrt{224}\)?
3. How would you handle cube roots or higher-order radicals?
4. What is the role of perfect squares in simplifying radicals?
5. How do you simplify the radical \(\sqrt{50}\)?

**Tip**: Always look for the largest perfect square factor inside the radical to simplify the expression easily.

Go step by step to reduce the radical. √224

Learn how to simplify the square root of 224 using prime factorization and radical simplification techniques. This guide covers core concepts of simplifying radicals for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation