Question

Which numbers are irrational? Select all that apply.

StartFraction 9 Over 15 EndFraction
StartRoot 18 EndRoot
StartRoot 9 EndRoot
StartRoot 169 EndRoot
StartRoot 78 EndRoot
Pi

Answer 1

To determine which numbers are irrational among the options provided, let's analyze each one:

1. **StartFraction 9 Over 15 EndFraction**: This can be simplified to \( \frac{3}{5} \), which is a rational number.

2. **StartRoot 18 EndRoot**: This is equal to \( \sqrt{18} \), which can be simplified to \( \sqrt{9 \times 2} = 3\sqrt{2} \). Since \( \sqrt{2} \) is irrational, \( 3\sqrt{2} \) is also irrational. Therefore, \( \sqrt{18} \) is irrational.

3. **StartRoot 9 EndRoot**: This equals \( \sqrt{9} = 3 \), which is a rational number.

4. **StartRoot 169 EndRoot**: This equals \( \sqrt{169} = 13 \), which is a rational number.

5. **StartRoot 78 EndRoot**: This is \( \sqrt{78} \). Since 78 cannot be expressed as a perfect square, \( \sqrt{78} \) is irrational.

6. **Pi**: The number \( \pi \) is a well-known irrational number.

Based on this analysis, the irrational numbers from the provided options are:

- StartRoot 18 EndRoot (√18)
- StartRoot 78 EndRoot (√78)
- Pi (π)

So, the selected irrational numbers are:

- StartRoot 18 EndRoot
- StartRoot 78 EndRoot
- Pi