To find the cube root of 225 step by step:

1. Understand the problem: You are asked to find the cube root of 225, which is written as: $\sqrt[3]{225}$ This means finding a number that, when multiplied by itself three times, gives 225.

2. Estimate the range: The cube root of 225 should lie between two integers whose cubes are close to 225. $5^3 = 125 \quad \text{and} \quad 6^3 = 216 \quad \text{and} \quad 7^3 = 343$ Since $216 < 225 < 343$, the cube root of 225 is between 6 and 7.

3. Use the average method for approximation: Let's improve our estimate by averaging. Start with 6.5 (between 6 and 7): $6.5^3 = 6.5 \times 6.5 \times 6.5 = 274.625$ This is larger than 225, so the cube root must be less than 6.5.

4. Try a smaller value: Let’s try 6.1: $6.1^3 = 6.1 \times 6.1 \times 6.1 = 226.981$ This is still slightly larger than 225, so the cube root must be less than 6.1.

5. Refine the estimate further: Let's try 6.05: $6.05^3 = 6.05 \times 6.05 \times 6.05 = 221.485$ This is smaller than 225, so the cube root must be between 6.05 and 6.1.

6. Conclude: After refinement, you can conclude that the cube root of 225 is approximately 6.08. You can use a calculator for more precision, but 6.08 is a good estimate.

Would you like more details or have any questions?

Related questions:

1. What is the cube root of 343?
2. How do you find cube roots using a calculator?
3. How can you solve for cube roots using logarithms?
4. Can the cube root of a negative number be calculated similarly?
5. How do you calculate higher-order roots (like the 4th root)?

Tip:

For faster cube root approximations, learning to use a scientific calculator’s cube root function can save time.