Question

irrational or rational
11/12*9/22
√5*8
3+√12
π+(-2π)
-3/8+3/5
√6/2+(-1/2)
√3*√12

Answer 1

To determine if each expression is rational or irrational, we need to evaluate them individually:

1. **\( \frac{11}{12} \times \frac{9}{22} \)**
- 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.**

2. **\( \sqrt{5} \times 8 \)**
- 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. **\( 3 + \sqrt{12} \)**
- 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.**

4. **\( \pi + (-2\pi) \)**
- This simplifies to:
\[
\pi - 2\pi = -\pi
\]
- Since \( \pi \) is irrational, \( -\pi \) is also irrational.
- Therefore, **irrational.**

5. **\( -\frac{3}{8} + \frac{3}{5} \)**
- 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.**

6. **\( \frac{\sqrt{6}}{2} + (-\frac{1}{2}) \)**
- 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.**

7. **\( \sqrt{3} \times \sqrt{12} \)**
- 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:
1. \( \frac{11}{12} \times \frac{9}{22} \) - rational
2. \( \sqrt{5} \times 8 \) - irrational
3. \( 3 + \sqrt{12} \) - irrational
4. \( \pi + (-2\pi) \) - irrational
5. \( -\frac{3}{8} + \frac{3}{5} \) - rational
6. \( \frac{\sqrt{6}}{2} + (-\frac{1}{2}) \) - irrational
7. \( \sqrt{3} \times \sqrt{12} \) - rational