To find the cube root of 1600 using prime factorization, follow these steps:

### Step 1: Prime Factorization of 1600

First, break 1600 into its prime factors:

\[
1600 \div 2 = 800 \\
800 \div 2 = 400 \\
400 \div 2 = 200 \\
200 \div 2 = 100 \\
100 \div 2 = 50 \\
50 \div 2 = 25 \\
25 \div 5 = 5 \\
5 \div 5 = 1
\]

The prime factorization of 1600 is:

\[
1600 = 2^6 \times 5^2
\]

### Step 2: Cube Root of Each Factor

To find the cube root, we need to divide the exponents of the prime factors by 3:

\[
\text{Cube root of } 1600 = \sqrt[3]{2^6 \times 5^2} = 2^{\frac{6}{3}} \times 5^{\frac{2}{3}} = 2^2 \times 5^{\frac{2}{3}}
\]

This simplifies to:

\[
\text{Cube root of } 1600 = 4 \times \sqrt[3]{25}
\]

### Final Answer

The cube root of 1600 is \( 4 \times \sqrt[3]{25} \).

Would you like more details or have any questions?

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Here are five related questions:
1. How do you find the square root of a number using prime factorization?
2. What is the difference between square roots and cube roots in terms of prime factorization?
3. Can you find the cube root of 500 using prime factorization?
4. How can you express the cube root of non-perfect cubes as a combination of integers and radicals?
5. Why is prime factorization useful in finding roots?

**Tip:** When dealing with cube roots, remember that the exponents of the prime factors must be divisible by 3 to get a perfect integer cube root.

Discover how to find the cube root of 1600 using prime factorization. Learn step-by-step with detailed explanations and examples. Suitable for high school students and anyone interested in mathematical methods for finding cube roots.

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