Let's evaluate each expression step by step and determine which results are rational numbers.
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\( \frac{10}{2} \cdot 3\sqrt{8} \)
- 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.
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\( 2 \cdot \pi \)
- The result is \( 2\pi \).
- Since \( \pi \) is irrational, \( 2\pi \) is also irrational.
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\( \frac{\sqrt{81}}{9} \cdot \frac{1}{3} \)
- 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.
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\( \sqrt{5} \cdot \sqrt{5} \)
- This simplifies to: \[ \sqrt{5} \cdot \sqrt{5} = 5 \]
- Since \( 5 \) is a rational number.
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\( \frac{2}{\sqrt{3}} \cdot 1 \)
- 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:
- \( \frac{1}{3} \)
- \( 5 \)