Question
Identify which condition is necessary for a number to be rational.(1 point)
It must not be negative.
It must be written as a fraction.
It must not be an endless decimal.
It must be able to be expressed as an integer divided by a nonzero integer
It must not be negative.
It must be written as a fraction.
It must not be an endless decimal.
It must be able to be expressed as an integer divided by a nonzero integer
Answers
GPT-4o mini
The necessary condition for a number to be rational is: **It must be able to be expressed as an integer divided by a nonzero integer.**
Which of the following is irrational?(1 point)
52.781654292 −7 837 −7 837
3.6363636363636363... √34
52.781654292 −7 837 −7 837
3.6363636363636363... √34
GPT-4o mini
The number that is irrational among the options you provided is **√34**.
The other options can be expressed as rational numbers:
- \(52.781654292\) is a finite decimal and thus rational.
- \(-7\) is an integer and therefore rational.
- \(-7\) is also an integer and is rational.
- \(3.6363636363636363...\) is a repeating decimal and thus rational.
In contrast, the square root of a non-perfect square, such as 34, is an irrational number.
The other options can be expressed as rational numbers:
- \(52.781654292\) is a finite decimal and thus rational.
- \(-7\) is an integer and therefore rational.
- \(-7\) is also an integer and is rational.
- \(3.6363636363636363...\) is a repeating decimal and thus rational.
In contrast, the square root of a non-perfect square, such as 34, is an irrational number.
Which of the following is irrational?(1 point)
52.781654292
−7 837
3.6363636363636363...
√34
52.781654292
−7 837
3.6363636363636363...
√34
GPT-4o mini
The irrational number among the options you provided is **√34**.
The other numbers are as follows:
- **52.781654292**: This is a finite decimal, and therefore it is rational.
- **−7**: This is an integer, and it is rational.
- **3.6363636363636363...**: This is a repeating decimal, which is also rational.
Hence, **√34** is the only irrational number listed.
The other numbers are as follows:
- **52.781654292**: This is a finite decimal, and therefore it is rational.
- **−7**: This is an integer, and it is rational.
- **3.6363636363636363...**: This is a repeating decimal, which is also rational.
Hence, **√34** is the only irrational number listed.
Which of the following is irrational?(1 point)
52.781654292 −7 837 −7 837
3.6363636363636363... √3/4
52.781654292 −7 837 −7 837
3.6363636363636363... √3/4
GPT-4o mini
The number that is irrational among the options you provided is **√3/4**.
Here's a breakdown of the options:
- **52.781654292**: This is a finite decimal, which means it is rational.
- **−7**: This is an integer, and therefore it is rational.
- **3.6363636363636363...**: This is a repeating decimal, which is also rational.
- **√3/4**: The square root of 3 is an irrational number. When you divide an irrational number by a rational number (4 in this case), it remains irrational.
Therefore, the answer is **√3/4**.
Here's a breakdown of the options:
- **52.781654292**: This is a finite decimal, which means it is rational.
- **−7**: This is an integer, and therefore it is rational.
- **3.6363636363636363...**: This is a repeating decimal, which is also rational.
- **√3/4**: The square root of 3 is an irrational number. When you divide an irrational number by a rational number (4 in this case), it remains irrational.
Therefore, the answer is **√3/4**.
Estimate the value of 6 × 2 6 × √2 . Which of the following choices is the best estimate?(1 point) – √
6.4
9.8
8.4
7.1
6.4
9.8
8.4
7.1
GPT-4o mini
To estimate the value of \(6 \times 2 + 6 \times \sqrt{2}\), we can simplify it as follows:
1. **Calculate \(6 \times 2\)**:
\[
6 \times 2 = 12
\]
2. **Estimate \(\sqrt{2}\)**:
\(\sqrt{2} \approx 1.414\). For estimation, we can round this to about \(1.4\).
3. **Calculate \(6 \times \sqrt{2}\)**:
\[
6 \times \sqrt{2} \approx 6 \times 1.4 = 8.4
\]
4. **Now combine the two results**:
\[
12 + 8.4 = 20.4
\]
However, it appears that the question might involve evaluating \(6 \times (2 + \sqrt{2})\), which may be a mistake, but typically follows what was originally stated.
1. **Calculate \(2 + \sqrt{2}\)**:
\[
2 + \sqrt{2} \approx 2 + 1.4 = 3.4
\]
2. **Now calculate \(6 \times (2 + \sqrt{2})\)**:
\[
6 \times 3.4 = 20.4
\]
If we're looking at it this way, and assuming we're still looking to select from provided choices, we can check the original intent.
However, from the given options (6.4, 9.8, 8.4, 7.1), we see they are significantly separated from 20.4, suggesting an expectation for grouping between \(6 \times 2\) and \(6 \times \sqrt{2}\).
Looking directly at estimates from original intent of evaluating simply against 6 and 2, we see:
- The second step confirms the returns around \(8.4\).
Thus, the best estimate from the choices provided for \(6 \times 2 + 6 \times \sqrt{2}\) is:
**8.4**.
1. **Calculate \(6 \times 2\)**:
\[
6 \times 2 = 12
\]
2. **Estimate \(\sqrt{2}\)**:
\(\sqrt{2} \approx 1.414\). For estimation, we can round this to about \(1.4\).
3. **Calculate \(6 \times \sqrt{2}\)**:
\[
6 \times \sqrt{2} \approx 6 \times 1.4 = 8.4
\]
4. **Now combine the two results**:
\[
12 + 8.4 = 20.4
\]
However, it appears that the question might involve evaluating \(6 \times (2 + \sqrt{2})\), which may be a mistake, but typically follows what was originally stated.
1. **Calculate \(2 + \sqrt{2}\)**:
\[
2 + \sqrt{2} \approx 2 + 1.4 = 3.4
\]
2. **Now calculate \(6 \times (2 + \sqrt{2})\)**:
\[
6 \times 3.4 = 20.4
\]
If we're looking at it this way, and assuming we're still looking to select from provided choices, we can check the original intent.
However, from the given options (6.4, 9.8, 8.4, 7.1), we see they are significantly separated from 20.4, suggesting an expectation for grouping between \(6 \times 2\) and \(6 \times \sqrt{2}\).
Looking directly at estimates from original intent of evaluating simply against 6 and 2, we see:
- The second step confirms the returns around \(8.4\).
Thus, the best estimate from the choices provided for \(6 \times 2 + 6 \times \sqrt{2}\) is:
**8.4**.
Which statement is true about the square root of 28?(1 point)
It is between 4 and 5, probably closer to 5.
It is between 5 and 6, probably closer to 6.
It is between 6 and 7, probably closer to 6.
It is between 5 and 6, probably closer to 5.
It is between 4 and 5, probably closer to 5.
It is between 5 and 6, probably closer to 6.
It is between 6 and 7, probably closer to 6.
It is between 5 and 6, probably closer to 5.
GPT-4o mini
To determine which statement is true about the square root of 28, we can estimate its value.
1. First, we know that \(5^2 = 25\) and \(6^2 = 36\). Therefore, \(\sqrt{28}\) is between 5 and 6.
2. Next, we can approximate:
- Since \(28\) is closer to \(25\) than it is to \(36\), \(\sqrt{28}\) is likely closer to \(5\).
Thus, the correct statement is:
**It is between 5 and 6, probably closer to 5.**
1. First, we know that \(5^2 = 25\) and \(6^2 = 36\). Therefore, \(\sqrt{28}\) is between 5 and 6.
2. Next, we can approximate:
- Since \(28\) is closer to \(25\) than it is to \(36\), \(\sqrt{28}\) is likely closer to \(5\).
Thus, the correct statement is:
**It is between 5 and 6, probably closer to 5.**