What is a rational number with a denominator of 7 that is between 7–√ and 8–√. Write your answer as an improper fraction.(1 point)
To find a rational number between $7-\sqrt{7}$ and $8-\sqrt{7},$ we must recognize that $\sqrt{7}$ is between $2$ and $3.$ In particular, $\sqrt{7} < 3.$
Therefore,
\[7 - \sqrt{7} < 7 - 3 = 4.\]Also,
\[8 - \sqrt{7} < 8 - 2 = 6.\]Thus, the rational number we seek must be between $\frac{4}{7}$ and $\frac{6}{7}.$ To find a fraction with denominator 7 between $\frac{4}{7}$ and $\frac{6}{7}$, we take
\[\frac{4}{7} + \frac{1}{7} = \boxed{\frac{5}{7}}.\]
To find a rational number between 7-√ and 8-√ with a denominator of 7, we need to find the rational number that is halfway between these two numbers.
Let's first find the halfway point between 7-√ and 8-√:
(7-√ + 8-√)/2
Simplifying this expression:
(15-2√)/2
Dividing both the numerator and denominator by 2:
(15/2 - 2√/2)
The rational number halfway between 7-√ and 8-√ is (15/2 - √2).
To write this answer as an improper fraction, we need to rationalize the denominator. To do this, we multiply the numerator and denominator by the conjugate of √2 which is (√2):
((15/2 - √2) * (√2))/(√2)
Simplifying the expression:
(15√2 - 2)/(√2)
To rationalize the denominator, we multiply the numerator and denominator by √2:
(15√2*√2 - 2*√2)/(√2*√2)
Simplifying the expression:
(30 - 2√2)/(2)
Now, we can write the answer as an improper fraction:
(30 - 2√2)/2 = (30/2 - 2√2/2) = 15 - √2
Therefore, the rational number with a denominator of 7 that is between 7-√ and 8-√ is written as an improper fraction as 15 - √2.