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Express 8/11 as a repeating decimal. Mark the cycle of repeating digits with a bar.(1 point)Question
Express 811
as a repeating decimal. Mark the cycle of repeating digits with a bar.(1 point)
$$
as a repeating decimal. Mark the cycle of repeating digits with a bar.(1 point)
$$
Answers
Answer
Express 8/11 as a repeating decimal. Mark the cycle of repeating digits with a bar.(1 point)
$$
$$
Answer
what
Answer
Convert 0.72¯¯¯¯¯
to a rational number in simplest form.(1 point)
to a rational number in simplest form.(1 point)
Answer
Which of the following is a rational number between 5 and 514
?(1 point)
Responses
517
5 Start Fraction 1 over 7 end fraction
513
5 Start Fraction 1 over 3 end fraction
26−−√
start root 26 end root
525
5 Start Fraction 2 over 5 end fraction
?(1 point)
Responses
517
5 Start Fraction 1 over 7 end fraction
513
5 Start Fraction 1 over 3 end fraction
26−−√
start root 26 end root
525
5 Start Fraction 2 over 5 end fraction
Answer
Which of the following is a rational number between 5 and 514
?(1 point)
Responses
5 1/7
5 1/3
start root 26 end root
5 2/5
?(1 point)
Responses
5 1/7
5 1/3
start root 26 end root
5 2/5
Answered by
GPT-4o mini
To express \( 811 \) as a repeating decimal, we need to first understand what the decimal representation of the fraction \( \frac{811}{1} \) looks like. Since 811 is an integer, it can be expressed as:
\[
811.000000\ldots
\]
However, because \( 811 \) has no fractional part, we can also represent it as a repeating decimal:
\[
811.\overline{0}
\]
This indicates that after the decimal point, the digit \( 0 \) repeats indefinitely. Therefore, the repeating decimal representation of \( 811 \) is:
\[
811.\overline{0}
\]
\[
811.000000\ldots
\]
However, because \( 811 \) has no fractional part, we can also represent it as a repeating decimal:
\[
811.\overline{0}
\]
This indicates that after the decimal point, the digit \( 0 \) repeats indefinitely. Therefore, the repeating decimal representation of \( 811 \) is:
\[
811.\overline{0}
\]
Answered by
GPT-4o mini
To express \( \frac{8}{11} \) as a repeating decimal, we can perform the division:
1. Divide \( 8.000000 \) by \( 11 \).
- \( 11 \) goes into \( 80 \) seven times (since \( 11 \times 7 = 77 \)).
- This leaves a remainder of \( 80 - 77 = 3 \).
2. Bring down a zero, making it \( 30 \).
- \( 11 \) goes into \( 30 \) twice (since \( 11 \times 2 = 22 \)).
- This leaves a remainder of \( 30 - 22 = 8 \).
3. Bring down another zero, making it \( 80 \) again.
- \( 11 \) goes into \( 80 \) seven times (as before).
- This leaves a remainder of \( 3 \) once more.
At this point, we see that we are going to cycle through the same steps infinitely: \( 7 \) followed by \( 2 \).
So, the decimal representation of \( \frac{8}{11} \) is:
\[
0.\overline{72}
\]
Thus, we mark the repeating cycle:
\[
\frac{8}{11} = 0.\overline{72}
\]
1. Divide \( 8.000000 \) by \( 11 \).
- \( 11 \) goes into \( 80 \) seven times (since \( 11 \times 7 = 77 \)).
- This leaves a remainder of \( 80 - 77 = 3 \).
2. Bring down a zero, making it \( 30 \).
- \( 11 \) goes into \( 30 \) twice (since \( 11 \times 2 = 22 \)).
- This leaves a remainder of \( 30 - 22 = 8 \).
3. Bring down another zero, making it \( 80 \) again.
- \( 11 \) goes into \( 80 \) seven times (as before).
- This leaves a remainder of \( 3 \) once more.
At this point, we see that we are going to cycle through the same steps infinitely: \( 7 \) followed by \( 2 \).
So, the decimal representation of \( \frac{8}{11} \) is:
\[
0.\overline{72}
\]
Thus, we mark the repeating cycle:
\[
\frac{8}{11} = 0.\overline{72}
\]
Answered by
GPT-4o mini
I apologize for any confusion. Let's clearly express \( \frac{8}{11} \) step-by-step as a repeating decimal.
1. **Perform long division** of \( 8 \) by \( 11 \):
- \( 11 \) goes into \( 8 \) zero times, so we start with \( 0.\)
- After placing the decimal, we consider \( 80 \) (after adding a zero).
- \( 11 \) goes into \( 80 \) seven times (since \( 11 \times 7 = 77 \)), leaving a remainder of \( 3 \) (i.e., \( 80 - 77 = 3 \)).
2. **Continue the division** by adding a zero to the remainder:
- Bring down a \( 0 \) to get \( 30 \).
- \( 11 \) goes into \( 30 \) two times (since \( 11 \times 2 = 22 \)), leaving a remainder of \( 8 \) (i.e., \( 30 - 22 = 8 \)).
3. **Recognize the cycle**:
- Now, we have the same remainder \( 8 \) again, and this brings us back to the beginning of our process.
- Next, we bring down another \( 0 \), yielding \( 80 \) again, continuing the cycle of \( 7 \) and \( 2 \).
Putting this all together, the decimal representation of \( \frac{8}{11} \) is:
\[
0.\overline{72}
\]
Thus, \( \frac{8}{11} \) expressed as a repeating decimal is:
\[
0.\overline{72}
\]
Thank you for your patience, and I hope this clarifies your question!
1. **Perform long division** of \( 8 \) by \( 11 \):
- \( 11 \) goes into \( 8 \) zero times, so we start with \( 0.\)
- After placing the decimal, we consider \( 80 \) (after adding a zero).
- \( 11 \) goes into \( 80 \) seven times (since \( 11 \times 7 = 77 \)), leaving a remainder of \( 3 \) (i.e., \( 80 - 77 = 3 \)).
2. **Continue the division** by adding a zero to the remainder:
- Bring down a \( 0 \) to get \( 30 \).
- \( 11 \) goes into \( 30 \) two times (since \( 11 \times 2 = 22 \)), leaving a remainder of \( 8 \) (i.e., \( 30 - 22 = 8 \)).
3. **Recognize the cycle**:
- Now, we have the same remainder \( 8 \) again, and this brings us back to the beginning of our process.
- Next, we bring down another \( 0 \), yielding \( 80 \) again, continuing the cycle of \( 7 \) and \( 2 \).
Putting this all together, the decimal representation of \( \frac{8}{11} \) is:
\[
0.\overline{72}
\]
Thus, \( \frac{8}{11} \) expressed as a repeating decimal is:
\[
0.\overline{72}
\]
Thank you for your patience, and I hope this clarifies your question!
Answered by
GPT-4o mini
To convert the repeating decimal \( 0.72\overline{72} \) into a rational number, we can follow these steps:
1. **Let \( x \) equal the repeating decimal**:
\[
x = 0.727272\ldots
\]
2. **Multiply \( x \) by \( 100 \)** (because the repeating part has 2 digits):
\[
100x = 72.727272\ldots
\]
3. **Set up an equation** by subtracting the first equation from the second:
\[
100x - x = 72.727272\ldots - 0.727272\ldots
\]
This simplifies to:
\[
99x = 72
\]
4. **Solve for \( x \)**:
\[
x = \frac{72}{99}
\]
5. **Simplify \( \frac{72}{99} \)**:
- Find the greatest common divisor (GCD) of \( 72 \) and \( 99 \). The GCD is \( 9 \).
- Divide the numerator and the denominator by \( 9 \):
\[
\frac{72 \div 9}{99 \div 9} = \frac{8}{11}
\]
Thus, the rational number equivalent of \( 0.72\overline{72} \) in simplest form is:
\[
\frac{8}{11}
\]
1. **Let \( x \) equal the repeating decimal**:
\[
x = 0.727272\ldots
\]
2. **Multiply \( x \) by \( 100 \)** (because the repeating part has 2 digits):
\[
100x = 72.727272\ldots
\]
3. **Set up an equation** by subtracting the first equation from the second:
\[
100x - x = 72.727272\ldots - 0.727272\ldots
\]
This simplifies to:
\[
99x = 72
\]
4. **Solve for \( x \)**:
\[
x = \frac{72}{99}
\]
5. **Simplify \( \frac{72}{99} \)**:
- Find the greatest common divisor (GCD) of \( 72 \) and \( 99 \). The GCD is \( 9 \).
- Divide the numerator and the denominator by \( 9 \):
\[
\frac{72 \div 9}{99 \div 9} = \frac{8}{11}
\]
Thus, the rational number equivalent of \( 0.72\overline{72} \) in simplest form is:
\[
\frac{8}{11}
\]
Answered by
GPT-4o mini
To determine which of the options provided is a rational number between 5 and 514, let's evaluate each option:
1. **517**: This number is greater than 514, so it does not fit in the range.
2. **\( 5 \frac{1}{7} \)**: This is a mixed number that can be converted to an improper fraction:
\[
5 \frac{1}{7} = 5 + \frac{1}{7} = \frac{35}{7} + \frac{1}{7} = \frac{36}{7} \approx 5.14
\]
This is greater than 5 and less than 514.
3. **513**: This number is less than 514 but greater than 5, so it fits into the range.
4. **\( 5 \frac{1}{3} \)**: This is another mixed number:
\[
5 \frac{1}{3} = 5 + \frac{1}{3} = \frac{15}{3} + \frac{1}{3} = \frac{16}{3} \approx 5.33
\]
This is also greater than 5 and less than 514.
5. **\( \sqrt{26} \)**: This number is approximately \( 5.099 \) (since \( 5^2 = 25 < 26 < 36 = 6^2 \)). This fits in the range between 5 and 514.
6. **525**: This number is greater than 514, so it does not fit in the range.
7. **\( 5 \frac{2}{5} \)**: This mixed number can be converted as follows:
\[
5 \frac{2}{5} = 5 + \frac{2}{5} = \frac{25}{5} + \frac{2}{5} = \frac{27}{5} \approx 5.4
\]
This is also greater than 5 and less than 514.
### Conclusion
The rational numbers between 5 and 514 from the options provided are:
- \( 5 \frac{1}{7} \) (approximately \( 5.14 \))
- \( 513 \)
- \( 5 \frac{1}{3} \) (approximately \( 5.33 \))
- \( \sqrt{26} \) (approximately \( 5.099 \))
- \( 5 \frac{2}{5} \) (approximately \( 5.4 \))
Thus, the rational numbers that fit the criteria are: **\( 5 \frac{1}{7} \), 513, \( 5 \frac{1}{3} \), \( \sqrt{26} \), and \( 5 \frac{2}{5} \)**.
Choose any of these to satisfy the condition of being a rational number between 5 and 514.
1. **517**: This number is greater than 514, so it does not fit in the range.
2. **\( 5 \frac{1}{7} \)**: This is a mixed number that can be converted to an improper fraction:
\[
5 \frac{1}{7} = 5 + \frac{1}{7} = \frac{35}{7} + \frac{1}{7} = \frac{36}{7} \approx 5.14
\]
This is greater than 5 and less than 514.
3. **513**: This number is less than 514 but greater than 5, so it fits into the range.
4. **\( 5 \frac{1}{3} \)**: This is another mixed number:
\[
5 \frac{1}{3} = 5 + \frac{1}{3} = \frac{15}{3} + \frac{1}{3} = \frac{16}{3} \approx 5.33
\]
This is also greater than 5 and less than 514.
5. **\( \sqrt{26} \)**: This number is approximately \( 5.099 \) (since \( 5^2 = 25 < 26 < 36 = 6^2 \)). This fits in the range between 5 and 514.
6. **525**: This number is greater than 514, so it does not fit in the range.
7. **\( 5 \frac{2}{5} \)**: This mixed number can be converted as follows:
\[
5 \frac{2}{5} = 5 + \frac{2}{5} = \frac{25}{5} + \frac{2}{5} = \frac{27}{5} \approx 5.4
\]
This is also greater than 5 and less than 514.
### Conclusion
The rational numbers between 5 and 514 from the options provided are:
- \( 5 \frac{1}{7} \) (approximately \( 5.14 \))
- \( 513 \)
- \( 5 \frac{1}{3} \) (approximately \( 5.33 \))
- \( \sqrt{26} \) (approximately \( 5.099 \))
- \( 5 \frac{2}{5} \) (approximately \( 5.4 \))
Thus, the rational numbers that fit the criteria are: **\( 5 \frac{1}{7} \), 513, \( 5 \frac{1}{3} \), \( \sqrt{26} \), and \( 5 \frac{2}{5} \)**.
Choose any of these to satisfy the condition of being a rational number between 5 and 514.
Answered by
GPT-4o mini
To determine which of the options provided is a rational number between 5 and 514, let's evaluate them one by one:
1. **\( 5 \frac{1}{7} \)**:
- This is a mixed number:
\[
5 \frac{1}{7} = 5 + \frac{1}{7} = \frac{35}{7} + \frac{1}{7} = \frac{36}{7} \approx 5.14
\]
- This value is greater than 5 and less than 514.
2. **\( 5 \frac{1}{3} \)**:
- This is a mixed number:
\[
5 \frac{1}{3} = 5 + \frac{1}{3} = \frac{15}{3} + \frac{1}{3} = \frac{16}{3} \approx 5.33
\]
- This value is also greater than 5 and less than 514.
3. **\( \sqrt{26} \)**:
- The square root of \( 26 \) is approximately \( 5.099 \) (since \( 5^2 = 25 < 26 < 36 = 6^2 \)).
- This value is greater than 5 and less than 514, but it is not a rational number.
4. **\( 5 \frac{2}{5} \)**:
- This is a mixed number:
\[
5 \frac{2}{5} = 5 + \frac{2}{5} = \frac{25}{5} + \frac{2}{5} = \frac{27}{5} \approx 5.4
\]
- This value is also greater than 5 and less than 514.
### Conclusion
The rational numbers between 5 and 514 from the options provided are:
- \( 5 \frac{1}{7} \) (approximately \( 5.14 \))
- \( 5 \frac{1}{3} \) (approximately \( 5.33 \))
- \( 5 \frac{2}{5} \) (approximately \( 5.4 \))
The only option that **is not a rational number** is \( \sqrt{26} \).
Thus, the correct responses that are rational numbers between 5 and 514 are:
- \( 5 \frac{1}{7} \)
- \( 5 \frac{1}{3} \)
- \( 5 \frac{2}{5} \)
Any of the first three options (excluding \( \sqrt{26} \)) qualify as rational numbers within the specified range.
1. **\( 5 \frac{1}{7} \)**:
- This is a mixed number:
\[
5 \frac{1}{7} = 5 + \frac{1}{7} = \frac{35}{7} + \frac{1}{7} = \frac{36}{7} \approx 5.14
\]
- This value is greater than 5 and less than 514.
2. **\( 5 \frac{1}{3} \)**:
- This is a mixed number:
\[
5 \frac{1}{3} = 5 + \frac{1}{3} = \frac{15}{3} + \frac{1}{3} = \frac{16}{3} \approx 5.33
\]
- This value is also greater than 5 and less than 514.
3. **\( \sqrt{26} \)**:
- The square root of \( 26 \) is approximately \( 5.099 \) (since \( 5^2 = 25 < 26 < 36 = 6^2 \)).
- This value is greater than 5 and less than 514, but it is not a rational number.
4. **\( 5 \frac{2}{5} \)**:
- This is a mixed number:
\[
5 \frac{2}{5} = 5 + \frac{2}{5} = \frac{25}{5} + \frac{2}{5} = \frac{27}{5} \approx 5.4
\]
- This value is also greater than 5 and less than 514.
### Conclusion
The rational numbers between 5 and 514 from the options provided are:
- \( 5 \frac{1}{7} \) (approximately \( 5.14 \))
- \( 5 \frac{1}{3} \) (approximately \( 5.33 \))
- \( 5 \frac{2}{5} \) (approximately \( 5.4 \))
The only option that **is not a rational number** is \( \sqrt{26} \).
Thus, the correct responses that are rational numbers between 5 and 514 are:
- \( 5 \frac{1}{7} \)
- \( 5 \frac{1}{3} \)
- \( 5 \frac{2}{5} \)
Any of the first three options (excluding \( \sqrt{26} \)) qualify as rational numbers within the specified range.
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