Homework 99 16/01/25 State whether each of, the following numbers are rational, lrvational, or neither. \( \begin{array}{llll}\text { a) } \sqrt[3]{33} & \text { b) } \sqrt{-2} & \text { e) } \frac{\sqrt{2}}{\sqrt{2}} & \text { d) } \sqrt[3]{27+1} \\ \text { o) } \pi & & \end{array} \)

To determine whether each of the given numbers is rational, irrational, or neither, we will analyze each one step by step. 1. **Rational Numbers**: These are numbers that can be expressed as the quotient of two integers (i.e., \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b \neq 0 \)). 2. **Irrational Numbers**: These are numbers that cannot be expressed as a simple fraction. Their decimal expansions are non-repeating and non-terminating. 3. **Neither**: This category includes numbers that do not fit into the above two categories, such as complex numbers. Now, let's analyze each number: ### a) \( \sqrt[3]{33} \) - The cube root of 33 is not a perfect cube, and it cannot be expressed as a fraction of two integers. - **Conclusion**: Irrational. ### b) \( \sqrt{-2} \) - The square root of a negative number is not a real number; it is an imaginary number. - **Conclusion**: Neither (it's complex). ### c) \( \frac{\sqrt{2}}{\sqrt{2}} \) - This simplifies to \( 1 \) (since any non-zero number divided by itself is 1). - **Conclusion**: Rational. ### d) \( \sqrt[3]{27+1} \) - This simplifies to \( \sqrt[3]{28} \). Since 28 is not a perfect cube, \( \sqrt[3]{28} \) cannot be expressed as a fraction of two integers. - **Conclusion**: Irrational. ### e) \( \pi \) - The number \( \pi \) is known to be an irrational number; it cannot be expressed as a fraction of two integers. - **Conclusion**: Irrational. ### Summary of Results: - a) \( \sqrt[3]{33} \): Irrational - b) \( \sqrt{-2} \): Neither - c) \( \frac{\sqrt{2}}{\sqrt{2}} \): Rational - d) \( \sqrt[3]{27+1} \): Irrational - e) \( \pi \): Irrational If you have any further questions or need additional explanations, feel free to ask!