What is a rational number with a denominator of 7 that is between √7 and √8. Write your answer as an improper fraction.

Answer 1

To find a rational number with a denominator of 7 that is between √7 and √8, we can start by finding the whole numbers that √7 and √8 are between.

Since 2^2 = 4, and 3^2 = 9, we know that √7 is between 2 and 3.

Since 2^2 = 4 and 3^2 = 9, we know that √8 is between 2 and 3 as well.

Since √7 and √8 are both between 2 and 3, a rational number with a denominator of 7 that is between them will have a whole number part of 2.

To find the fractional part, we can compare the difference between √7 and √8 to the difference between 2 and 3.

The difference between √8 and √7 is √8 - √7 = √(8) - √(7) = √(4*2) - √(7) = 2√2 - √7.
The difference between 2 and 3 is 3 - 2 = 1.

Therefore, the fractional part of the rational number we are looking for is (2√2 - √7) / 1.

To rationalize the denominator, we can multiply the numerator and denominator by the conjugate of the denominator.
The conjugate of 1 is 1, so we multiply both the numerator and denominator by 1.

((2√2 - √7) / 1) * (1 / 1) = (2√2 - √7) / 1 = 2√2 - √7

Therefore, the rational number we are looking for is 2√2 - √7, written as an improper fraction.

Answer 2

could you say this simpler

Answer 3

Sure! A simpler way to express a rational number with a denominator of 7 between √7 and √8 is to start by recognizing that both √7 and √8 are between the whole numbers 2 and 3. Since we want the rational number to have a denominator of 7, we can choose the whole number part to be 2.

Next, we look at the fractional part. The difference between √8 and √7 can be written as √8 - √7 = √(8) - √(7) = √(4*2) - √(7) = 2√2 - √7.

Finally, we can write this as an improper fraction by keeping the numerator the same and putting it over 1 (since we want a denominator of 7). So the rational number is (2√2 - √7) / 1 = 2√2 - √7.