Question

Solve and show which result is a rational number.
10/2 • 3√8
2 • pi
√81/9 • 1/3
√5 • √5
2/√3 • 1

Answer 1

Let's evaluate each expression step by step and determine which results are rational numbers.

1. **\( \frac{10}{2} \cdot 3\sqrt{8} \)**

- First, simplify \( \frac{10}{2} = 5 \).
- Next, simplify \( \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \).
- Now substitute back:
\[
5 \cdot 3(2\sqrt{2}) = 5 \cdot 6\sqrt{2} = 30\sqrt{2}
\]
- Since \( \sqrt{2} \) is irrational, \( 30\sqrt{2} \) is also irrational.

2. **\( 2 \cdot \pi \)**

- The result is \( 2\pi \).
- Since \( \pi \) is irrational, \( 2\pi \) is also irrational.

3. **\( \frac{\sqrt{81}}{9} \cdot \frac{1}{3} \)**

- First, calculate \( \sqrt{81} = 9 \).
- Now substitute:
\[
\frac{9}{9} \cdot \frac{1}{3} = 1 \cdot \frac{1}{3} = \frac{1}{3}
\]
- Since \( \frac{1}{3} \) is a rational number.

4. **\( \sqrt{5} \cdot \sqrt{5} \)**

- This simplifies to:
\[
\sqrt{5} \cdot \sqrt{5} = 5
\]
- Since \( 5 \) is a rational number.

5. **\( \frac{2}{\sqrt{3}} \cdot 1 \)**

- The result stays:
\[
\frac{2}{\sqrt{3}}
\]
- This is not rational since \( \sqrt{3} \) is irrational, making \( \frac{2}{\sqrt{3}} \) irrational.

### Summary of Results:
- \( \frac{10}{2} \cdot 3\sqrt{8} = 30\sqrt{2} \) (irrational)
- \( 2 \cdot \pi = 2\pi \) (irrational)
- \( \frac{\sqrt{81}}{9} \cdot \frac{1}{3} = \frac{1}{3} \) (rational)
- \( \sqrt{5} \cdot \sqrt{5} = 5 \) (rational)
- \( \frac{2}{\sqrt{3}} \cdot 1 = \frac{2}{\sqrt{3}} \) (irrational)

The rational results are:
- \( \frac{1}{3} \)
- \( 5 \)