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The problem asks to simplify the radical \(\sqrt{176}\) step by step. Here's how we can approach this:

### Step 1: Prime Factorization of 176
First, let's factor 176 into prime factors:
\[
176 \div 2 = 88
\]
\[
88 \div 2 = 44
\]
\[
44 \div 2 = 22
\]
\[
22 \div 2 = 11
\]
\[
11 \div 11 = 1
\]

So, the prime factorization of 176 is:
\[
176 = 2^4 \times 11
\]

### Step 2: Simplifying the Square Root
We use the property of square roots: \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\).

Now simplify \(\sqrt{176}\):
\[
\sqrt{176} = \sqrt{2^4 \times 11} = \sqrt{2^4} \times \sqrt{11}
\]

Since \(\sqrt{2^4} = 2^2 = 4\), we have:
\[
\sqrt{176} = 4\sqrt{11}
\]

### Final Answer:
The simplified form of \(\sqrt{176}\) is \(4\sqrt{11}\).

Would you like more details on any of these steps?

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Here are 5 related questions:
1. How do you find the prime factors of a number?
2. What are the properties of square roots used in simplification?
3. How do you handle square roots of non-perfect squares?
4. What is the prime factorization of another number, such as 180?
5. Can all radicals be simplified? Why or why not?

**Tip:** Always look for perfect squares within the radical to simplify it efficiently!

Go step by step to reduce the radical √176

Learn how to simplify the radical √176 by breaking it down using prime factorization and square root properties. This detailed solution explains how to reduce the square root to 4√11, suitable for students in Grades 8-10.

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