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

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

UpStudy AI · Accepted Answer

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 $$ b 
eq 0 $$).

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!
- 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

Homework 99
16/01/25
State whether each of, the following numbers
are rational, lrvational, or neither.

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