To solve the expression \( \sqrt[3]{420} \), which represents the cube root of 420, follow these steps:

### Steps:

1. **Estimate the cube root**:
   The cube root of a number \( n \) is a value \( x \) such that \( x^3 = n \). In this case, we need to find a value of \( x \) such that \( x^3 = 420 \).

2. **Check nearby perfect cubes**:
   - \( 7^3 = 343 \)
   - \( 8^3 = 512 \)

   Since \( 420 \) lies between 343 and 512, the cube root of 420 will be between 7 and 8.

3. **Refine using approximation**:
   The value of \( \sqrt[3]{420} \) is approximately 7.5. You can get a more precise value using a calculator:

   \[
   \sqrt[3]{420} \approx 7.497
   \]

Thus, the cube root of 420 is approximately **7.497**.

Would you like more details on this method or a deeper explanation?

Here are five related questions:
1. How do you find cube roots without a calculator?
2. What are perfect cubes, and how can they help with cube root estimation?
3. What’s the difference between square roots and cube roots?
4. How can we use Newton’s method to approximate cube roots?
5. What is the cube root of a negative number?

**Tip:** Memorizing common cubes like \( 1^3 = 1 \), \( 2^3 = 8 \), \( 3^3 = 27 \), etc., can help estimate cube roots quickly.

Learn how to solve the cube root of 420 with a detailed, step-by-step explanation. This problem covers key concepts in algebra, including cube roots, estimation, and the cube root theorem. Suitable for students in Grades 8-10.

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