Question

14

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

(1 point)

Answer 1

To find a rational number with a denominator of 7 that is between √(7) and √(8), we can simply take the average of those two numbers:

(√(7) + √(8)) / 2.

To write this as an improper fraction, we need to rationalize the denominator by multiplying the top and bottom by the conjugate of the denominator:

(√(7) + √(8)) / 2 = (√(7) + √(8)) / 2 * (√(7) - √(8)) / (√(7) - √(8)).

Expanding the numerator and denominator:

(√(49) - √(7*8) + √(7*8) - √(64))/(2*√(7) - 2*√(8)).

Simplifying:

(√(49) - √(64))/(2*√(7) - 2*√(8)).

Which further simplifies to:

(7 - 8) / (2√(7) - 2√(8)).

So, the rational number with a denominator of 7 that is between √(7) and √(8) is -1 / (2√(7) - 2√(8)).