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Let's solve each problem step-by-step:
### Q1: Insert three rational numbers between [tex]\(-\frac{2}{3}\)[/tex] and [tex]\(-\frac{1}{3}\)[/tex].
To find three rational numbers between [tex]\(-\frac{2}{3}\)[/tex] and [tex]\(-\frac{1}{3}\)[/tex]:
1. The first rational number can be [tex]\(-\frac{2}{3} + \frac{1}{12} = -0.5833333333333333\)[/tex].
2. The second rational number can be [tex]\(-\frac{2}{3} + \frac{1}{6} = -0.5\)[/tex].
3. The third rational number can be [tex]\(-\frac{2}{3} + \frac{1}{4} = -0.41666666666666663\)[/tex].
So, the three rational numbers are [tex]\( -0.5833333333333333 \)[/tex], [tex]\( -0.5 \)[/tex], and [tex]\( -0.41666666666666663 \)[/tex].
### Q2: Calculate the [tex]\( \frac{p}{q} \)[/tex] form of [tex]\(0.777...\)[/tex], where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex].
The repeating decimal [tex]\(0.777...\)[/tex] can be expressed as a fraction. It is equal to [tex]\(\frac{7}{9}\)[/tex].
So, [tex]\(p = 7\)[/tex] and [tex]\(q = 9\)[/tex].
### Q3: Write the sum of [tex]\(0.\overline{3}\)[/tex] and 0.4.
To find the sum of [tex]\(0.\overline{3}\)[/tex] and [tex]\(0.4\)[/tex]:
- [tex]\(0.\overline{3}\)[/tex] as a fraction is [tex]\( \frac{1}{3} \)[/tex].
- Adding [tex]\(0.4\)[/tex] gives us [tex]\( \frac{1}{3} + 0.4 \)[/tex].
The sum is approximately [tex]\(0.7333333333333334\)[/tex].
### Q4: Express [tex]\(0.32\overline{28}\)[/tex] in the form of [tex]\(\frac{p}{q}\)[/tex], where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex].
To express [tex]\(0.32\overline{28}\)[/tex] in fraction form, follow these steps:
1. Let [tex]\(x = 0.322828...\)[/tex].
2. Multiply by 1000: [tex]\(1000x = 322.82828...\)[/tex].
3. Multiply by 10: [tex]\(10x = 3.2282828...\)[/tex].
Subtract these equations:
[tex]\[1000x - 10x = 322.82828... - 3.2282828...\][/tex]
[tex]\[990x = 322.5\][/tex]
Thus, [tex]\( x = \frac{322.5}{990} \)[/tex].
### Q5: Express [tex]\(0.6 + 0.7 + 0.4\overline{7}\)[/tex] in the form of [tex]\( \frac{p}{q}\)[/tex].
To find the sum of [tex]\(0.6 + 0.7 + 0.4\overline{7}\)[/tex]:
1. [tex]\(0.6 + 0.7 = 1.3\)[/tex].
2. The repeating decimal [tex]\(0.4\overline{7}\)[/tex] as a fraction is [tex]\(0.4 + \frac{4}{9} = 0.877777...\)[/tex].
The sum is approximately [tex]\(2.1444444444444444\)[/tex].
### Q6: Calculate the irrational number between [tex]\(2\)[/tex] and [tex]\(2.5\)[/tex].
An example of an irrational number between [tex]\(2\)[/tex] and [tex]\(2.5\)[/tex] is approximately [tex]\(2.3067071956034932\)[/tex].
### Q7: Simplify [tex]\(3 \sqrt{45} - \sqrt{125} + \sqrt{200} - \sqrt{50}\)[/tex].
To simplify [tex]\(3 \sqrt{45} - \sqrt{125} + \sqrt{200} - \sqrt{50}\)[/tex]:
1. [tex]\(3 \sqrt{45} = 3 \cdot 3 \sqrt{5} = 9 \sqrt{5}\)[/tex].
2. [tex]\(\sqrt{125} = 5 \sqrt{5}\)[/tex].
3. [tex]\(\sqrt{200} = \sqrt{4 \cdot 50} = 2 \cdot \sqrt{50} = 2 \cdot 5 \sqrt{2} = 10 \sqrt{2}\)[/tex].
4. [tex]\(\sqrt{50} = 5 \sqrt{2}\)[/tex].
Now, combine them:
[tex]\[3 \sqrt{45} - \sqrt{125} + \sqrt{200} - \sqrt{50} = 9 \sqrt{5} - 5 \sqrt{5} + 10 \sqrt{2} - 5 \sqrt{2}\][/tex]
[tex]\[= (9 \sqrt{5} - 5 \sqrt{5}) + (10 \sqrt{2} - 5 \sqrt{2})\][/tex]
[tex]\[= 4 \sqrt{5} + 5 \sqrt{2}\][/tex]
And hence, the simplified form is [tex]\(5 \sqrt{2} + 4 \sqrt{5}\)[/tex].
### Q1: Insert three rational numbers between [tex]\(-\frac{2}{3}\)[/tex] and [tex]\(-\frac{1}{3}\)[/tex].
To find three rational numbers between [tex]\(-\frac{2}{3}\)[/tex] and [tex]\(-\frac{1}{3}\)[/tex]:
1. The first rational number can be [tex]\(-\frac{2}{3} + \frac{1}{12} = -0.5833333333333333\)[/tex].
2. The second rational number can be [tex]\(-\frac{2}{3} + \frac{1}{6} = -0.5\)[/tex].
3. The third rational number can be [tex]\(-\frac{2}{3} + \frac{1}{4} = -0.41666666666666663\)[/tex].
So, the three rational numbers are [tex]\( -0.5833333333333333 \)[/tex], [tex]\( -0.5 \)[/tex], and [tex]\( -0.41666666666666663 \)[/tex].
### Q2: Calculate the [tex]\( \frac{p}{q} \)[/tex] form of [tex]\(0.777...\)[/tex], where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex].
The repeating decimal [tex]\(0.777...\)[/tex] can be expressed as a fraction. It is equal to [tex]\(\frac{7}{9}\)[/tex].
So, [tex]\(p = 7\)[/tex] and [tex]\(q = 9\)[/tex].
### Q3: Write the sum of [tex]\(0.\overline{3}\)[/tex] and 0.4.
To find the sum of [tex]\(0.\overline{3}\)[/tex] and [tex]\(0.4\)[/tex]:
- [tex]\(0.\overline{3}\)[/tex] as a fraction is [tex]\( \frac{1}{3} \)[/tex].
- Adding [tex]\(0.4\)[/tex] gives us [tex]\( \frac{1}{3} + 0.4 \)[/tex].
The sum is approximately [tex]\(0.7333333333333334\)[/tex].
### Q4: Express [tex]\(0.32\overline{28}\)[/tex] in the form of [tex]\(\frac{p}{q}\)[/tex], where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex].
To express [tex]\(0.32\overline{28}\)[/tex] in fraction form, follow these steps:
1. Let [tex]\(x = 0.322828...\)[/tex].
2. Multiply by 1000: [tex]\(1000x = 322.82828...\)[/tex].
3. Multiply by 10: [tex]\(10x = 3.2282828...\)[/tex].
Subtract these equations:
[tex]\[1000x - 10x = 322.82828... - 3.2282828...\][/tex]
[tex]\[990x = 322.5\][/tex]
Thus, [tex]\( x = \frac{322.5}{990} \)[/tex].
### Q5: Express [tex]\(0.6 + 0.7 + 0.4\overline{7}\)[/tex] in the form of [tex]\( \frac{p}{q}\)[/tex].
To find the sum of [tex]\(0.6 + 0.7 + 0.4\overline{7}\)[/tex]:
1. [tex]\(0.6 + 0.7 = 1.3\)[/tex].
2. The repeating decimal [tex]\(0.4\overline{7}\)[/tex] as a fraction is [tex]\(0.4 + \frac{4}{9} = 0.877777...\)[/tex].
The sum is approximately [tex]\(2.1444444444444444\)[/tex].
### Q6: Calculate the irrational number between [tex]\(2\)[/tex] and [tex]\(2.5\)[/tex].
An example of an irrational number between [tex]\(2\)[/tex] and [tex]\(2.5\)[/tex] is approximately [tex]\(2.3067071956034932\)[/tex].
### Q7: Simplify [tex]\(3 \sqrt{45} - \sqrt{125} + \sqrt{200} - \sqrt{50}\)[/tex].
To simplify [tex]\(3 \sqrt{45} - \sqrt{125} + \sqrt{200} - \sqrt{50}\)[/tex]:
1. [tex]\(3 \sqrt{45} = 3 \cdot 3 \sqrt{5} = 9 \sqrt{5}\)[/tex].
2. [tex]\(\sqrt{125} = 5 \sqrt{5}\)[/tex].
3. [tex]\(\sqrt{200} = \sqrt{4 \cdot 50} = 2 \cdot \sqrt{50} = 2 \cdot 5 \sqrt{2} = 10 \sqrt{2}\)[/tex].
4. [tex]\(\sqrt{50} = 5 \sqrt{2}\)[/tex].
Now, combine them:
[tex]\[3 \sqrt{45} - \sqrt{125} + \sqrt{200} - \sqrt{50} = 9 \sqrt{5} - 5 \sqrt{5} + 10 \sqrt{2} - 5 \sqrt{2}\][/tex]
[tex]\[= (9 \sqrt{5} - 5 \sqrt{5}) + (10 \sqrt{2} - 5 \sqrt{2})\][/tex]
[tex]\[= 4 \sqrt{5} + 5 \sqrt{2}\][/tex]
And hence, the simplified form is [tex]\(5 \sqrt{2} + 4 \sqrt{5}\)[/tex].
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