What is a rational number with a denominator of 7 that is between the square root of 7 and the square root of 8. Write your answer as an improper fraction.
To begin, let's calculate the square roots of 7 and 8:
√7 ≈ 2.646...
√8 ≈ 2.828...
Now, we need to find a rational number with a denominator of 7 that lies between these two values. We know that √8 is greater than √7, so we need to find a rational number that is closer to √7 than √8.
To simplify the calculation, let's consider the decimal representation.
2.646... < x < 2.828...
To find a rational number between these two values, we could use the average:
(2.646... + 2.828...) / 2 = 2.738...
Now, we need to write this as an improper fraction with a denominator of 7.
To do this, multiply both the numerator and denominator by 7 to maintain the same value:
2.738... * 7 = 19.166...
Now, we can write this as the improper fraction:
19.166... / 7
To find the numerator, subtract the whole number part from the decimal:
19.166... - 19 = 0.166...
Now, place the numerator over the denominator to get:
0.166... / 7
Since the decimal repeats indefinitely, we can write it as a fraction:
0.166... = 1/6
Therefore, the rational number with a denominator of 7 that lies between √7 and √8 is 1/6.
So the answer is: 1/6.
To find a rational number with a denominator of 7 between the square root of 7 and the square root of 8, we can start by finding the square root of 7 and the square root of 8.
The square root of 7 is approximately 2.645751311.
The square root of 8 is approximately 2.828427125.
Now, we want to find a rational number with a denominator of 7 between these two values. Let's look for a number that falls between 2.645751311 and 2.828427125.
A good strategy is to find the average of these two numbers:
(2.645751311 + 2.828427125) / 2 = 2.737089218
To convert this to an improper fraction, we multiply both the numerator and denominator by 7:
2.737089218 * 7 = 19.16062453
So, the rational number with a denominator of 7 between the square root of 7 and the square root of 8 is approximately 19.16062453/7 as an improper fraction.