Question

Are the following numbers rational or irrational?
square root of 16
pi (3.14)
2.75
square root of 2
square root of 8
0.0333333...
^3square root of 8

Answer 1

To determine whether the numbers you've listed are rational or irrational, we can use the definitions:

- A **rational number** can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
- An **irrational number** cannot be expressed as a simple fraction.

Now let's analyze each number:

1. **Square root of 16**:
- \(\sqrt{16} = 4\), which is a whole number and thus rational.

2. **Pi (3.14)**:
- \(\pi\) is an irrational number because it cannot be expressed as a fraction. (Note: 3.14 is a rational approximation of \(\pi\), but \(\pi\) itself is irrational.)

3. **2.75**:
- This is a decimal that can be expressed as \(\frac{11}{4}\), which is a fraction. Therefore, it is rational.

4. **Square root of 2**:
- \(\sqrt{2}\) is known to be an irrational number.

5. **Square root of 8**:
- \(\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}\). Since \(\sqrt{2}\) is irrational, \(2\sqrt{2}\) is also irrational. Therefore, \(\sqrt{8}\) is irrational.

6. **0.0333333...**:
- This is a repeating decimal, which can be expressed as \(\frac{1}{30}\). Thus, it is a rational number.

7. **Cube root of 8**:
- \(\sqrt[3]{8} = 2\), which is a whole number and therefore rational.

In summary:
- Square root of 16: **Rational**
- Pi (3.14): **Irrational**
- 2.75: **Rational**
- Square root of 2: **Irrational**
- Square root of 8: **Irrational**
- 0.0333333...: **Rational**
- Cube root of 8: **Rational**