Question
is the number V49 rational with a terminating decimal
Answers
Answered by
GPT-4o mini
To determine if the number \( V49 \) (assuming \( V \) is the square root sign) is rational with a terminating decimal, we first need to interpret \( V49 \).
If \( V49 \) represents \( \sqrt{49} \), then:
\[
\sqrt{49} = 7
\]
The number 7 is a rational number. A rational number is one that can be expressed as a fraction of two integers (in this case, \( \frac{7}{1} \)).
Furthermore, the decimal representation of 7 is 7.0, which is a terminating decimal.
Thus, **yes**, the number \( V49 \) (or \( \sqrt{49} \)) is rational and has a terminating decimal.
If \( V49 \) represents \( \sqrt{49} \), then:
\[
\sqrt{49} = 7
\]
The number 7 is a rational number. A rational number is one that can be expressed as a fraction of two integers (in this case, \( \frac{7}{1} \)).
Furthermore, the decimal representation of 7 is 7.0, which is a terminating decimal.
Thus, **yes**, the number \( V49 \) (or \( \sqrt{49} \)) is rational and has a terminating decimal.
Answer
what about 33/8
Answered by
GPT-4o mini
To determine if the number \( \frac{33}{8} \) is rational and whether it has a terminating decimal, we first note that:
1. **Rationality**: A number is rational if it can be expressed as a fraction of two integers. Since \( 33 \) and \( 8 \) are both integers, \( \frac{33}{8} \) is indeed a rational number.
2. **Terminating Decimal**: A fraction in simplest form has a terminating decimal if the denominator (after simplification) can be expressed in the form \( 2^m \times 5^n \), where \( m \) and \( n \) are non-negative integers.
In the case of \( \frac{33}{8} \):
- The denominator 8 can be expressed as \( 2^3 \) (since \( 8 = 2 \times 2 \times 2 \)).
- Since there are no other prime factors in the denominator other than 2, \( \frac{33}{8} \) has a terminating decimal.
To find the decimal form:
\[
\frac{33}{8} = 4.125
\]
So, \( \frac{33}{8} \) is a rational number, and its decimal representation \( 4.125 \) is a terminating decimal.
1. **Rationality**: A number is rational if it can be expressed as a fraction of two integers. Since \( 33 \) and \( 8 \) are both integers, \( \frac{33}{8} \) is indeed a rational number.
2. **Terminating Decimal**: A fraction in simplest form has a terminating decimal if the denominator (after simplification) can be expressed in the form \( 2^m \times 5^n \), where \( m \) and \( n \) are non-negative integers.
In the case of \( \frac{33}{8} \):
- The denominator 8 can be expressed as \( 2^3 \) (since \( 8 = 2 \times 2 \times 2 \)).
- Since there are no other prime factors in the denominator other than 2, \( \frac{33}{8} \) has a terminating decimal.
To find the decimal form:
\[
\frac{33}{8} = 4.125
\]
So, \( \frac{33}{8} \) is a rational number, and its decimal representation \( 4.125 \) is a terminating decimal.
Answer
so yes?
Answered by
GPT-4o mini
Yes, the number \( \frac{33}{8} \) is a rational number and has a terminating decimal (specifically, \( 4.125 \)).
Answer
what about 1.3
Answered by
GPT-4o mini
The number \( 1.3 \) is a rational number, as it can be expressed as a fraction of two integers. Specifically:
\[
1.3 = \frac{13}{10}
\]
Since both \( 13 \) and \( 10 \) are integers, \( 1.3 \) is indeed rational.
Additionally, the decimal \( 1.3 \) is terminating because it has a finite number of digits after the decimal point.
So, in summary:
Yes, \( 1.3 \) is a rational number and has a terminating decimal.
\[
1.3 = \frac{13}{10}
\]
Since both \( 13 \) and \( 10 \) are integers, \( 1.3 \) is indeed rational.
Additionally, the decimal \( 1.3 \) is terminating because it has a finite number of digits after the decimal point.
So, in summary:
Yes, \( 1.3 \) is a rational number and has a terminating decimal.
Answer
2/11
Answered by
GPT-4o mini
To determine the characteristics of the number \( \frac{2}{11} \):
1. **Rationality**: \( \frac{2}{11} \) is a rational number because it can be expressed as a fraction of two integers (2 and 11).
2. **Terminating Decimal**: A fraction has a terminating decimal if, after simplification, the denominator (when in lowest terms) only contains the prime factors 2 and/or 5.
In the case of \( \frac{2}{11} \):
- The denominator 11 is a prime number and contains factors other than 2 and 5.
Therefore, \( \frac{2}{11} \) does **not** have a terminating decimal. Instead, it has a repeating decimal representation:
\[
\frac{2}{11} \approx 0.181818\ldots
\]
This can be written as \( 0.\overline{18} \), indicating that the digits "18" repeat indefinitely.
In summary:
- \( \frac{2}{11} \) is a rational number, but it does **not** have a terminating decimal; it has a repeating decimal.
1. **Rationality**: \( \frac{2}{11} \) is a rational number because it can be expressed as a fraction of two integers (2 and 11).
2. **Terminating Decimal**: A fraction has a terminating decimal if, after simplification, the denominator (when in lowest terms) only contains the prime factors 2 and/or 5.
In the case of \( \frac{2}{11} \):
- The denominator 11 is a prime number and contains factors other than 2 and 5.
Therefore, \( \frac{2}{11} \) does **not** have a terminating decimal. Instead, it has a repeating decimal representation:
\[
\frac{2}{11} \approx 0.181818\ldots
\]
This can be written as \( 0.\overline{18} \), indicating that the digits "18" repeat indefinitely.
In summary:
- \( \frac{2}{11} \) is a rational number, but it does **not** have a terminating decimal; it has a repeating decimal.