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Let's analyze the given problem.

**Question:**
Determine whether the root is a rational or irrational number. Explain your reasoning.

### Part (a): \( \sqrt{62} \)

To determine whether \( \sqrt{62} \) is rational or irrational, we need to check if 62 is a perfect square. A number is a perfect square if it can be expressed as the square of an integer.

- The square root of 62 is not an integer since 62 is not a perfect square (it is not the square of any integer).
- Hence, \( \sqrt{62} \) cannot be expressed as a fraction of two integers, making it an **irrational number**.

**Explanation:**
\( \sqrt{62} \) is irrational because 62 is not a perfect square, and its square root will result in a non-terminating, non-repeating decimal.

### Part (b): \( \sqrt[3]{100} \)

Now let's determine whether \( \sqrt[3]{100} \) is rational or irrational.

- The cube root of 100 is not an integer, as 100 is not a perfect cube (it is not the cube of any integer).
- Therefore, \( \sqrt[3]{100} \) cannot be written as a fraction of two integers, so it is also an **irrational number**.

**Explanation:**
\( \sqrt[3]{100} \) is irrational because 100 is not a perfect cube, and its cube root results in a non-terminating, non-repeating decimal.

Would you like any further details or explanations? Here are some additional related questions:

1. What are the characteristics of rational numbers?
2. How do you determine if a square root is rational or irrational?
3. What is the difference between a perfect square and a perfect cube?
4. Can a cube root of a non-perfect cube ever be rational?
5. How do decimal expansions of rational and irrational numbers differ?

**Tip:** For any number that is not a perfect square or perfect cube, its respective square root or cube root will always be irrational.

Determine whether the root is a rational or irrational number. Explain your reasoning for √62 and ³√100.

Explore whether √62 and ³√100 are rational or irrational numbers with detailed explanations. Understand how to identify perfect squares and perfect cubes. This math problem is ideal for students in Grades 8-10.

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