To determine if each expression is rational or irrational, we need to evaluate them individually:
-
\( \frac{11}{12} \times \frac{9}{22} \)
- Calculate: \[ \frac{11 \times 9}{12 \times 22} = \frac{99}{264} \]
- Simplify: \[ \frac{99 \div 99}{264 \div 99} = \frac{1}{2.6667} \approx \frac{1}{2.6667} \text{, which is rational.} \]
- Therefore, rational.
-
\( \sqrt{5} \times 8 \)
- This is \( 8\sqrt{5} \), and since \( \sqrt{5} \) is irrational, multiplying it by 8 (a rational number) results in an irrational number.
- Therefore, irrational.
-
\( 3 + \sqrt{12} \)
- Simplifying \( \sqrt{12} \): \[ \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \]
- Thus, \( 3 + 2\sqrt{3} \) is the sum of a rational number and an irrational number, which is irrational.
- Therefore, irrational.
-
\( \pi + (-2\pi) \)
- This simplifies to: \[ \pi - 2\pi = -\pi \]
- Since \( \pi \) is irrational, \( -\pi \) is also irrational.
- Therefore, irrational.
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\( -\frac{3}{8} + \frac{3}{5} \)
- To add these fractions, find a common denominator (40): \[ -\frac{3}{8} = -\frac{15}{40},\quad \frac{3}{5} = \frac{24}{40} \]
- Adding: \[ -\frac{15}{40} + \frac{24}{40} = \frac{9}{40} \]
- This result is rational.
- Therefore, rational.
-
\( \frac{\sqrt{6}}{2} + (-\frac{1}{2}) \)
- This can be rewritten as: \[ \frac{\sqrt{6}}{2} - \frac{1}{2} = \frac{\sqrt{6} - 1}{2} \]
- Since \( \sqrt{6} \) is irrational, \( \sqrt{6} - 1 \) is still irrational, and dividing an irrational number by a non-zero rational number results in an irrational number.
- Therefore, irrational.
-
\( \sqrt{3} \times \sqrt{12} \)
- This can be simplified: \[ \sqrt{3} \times \sqrt{12} = \sqrt{3 \times 12} = \sqrt{36} = 6 \]
- Since 6 is a rational number, this expression is rational.
- Therefore, rational.
To summarize:
- \( \frac{11}{12} \times \frac{9}{22} \) - rational
- \( \sqrt{5} \times 8 \) - irrational
- \( 3 + \sqrt{12} \) - irrational
- \( \pi + (-2\pi) \) - irrational
- \( -\frac{3}{8} + \frac{3}{5} \) - rational
- \( \frac{\sqrt{6}}{2} + (-\frac{1}{2}) \) - irrational
- \( \sqrt{3} \times \sqrt{12} \) - rational