What is a rational number with the denominator of 7 that is between square root of 7 and square root of 8. Write your answer as an improper fraction. Help
To find a rational number with a denominator of 7 that is between √7 and √8, we need to find a number that lies between these two values and can be written in the form of a fraction with a denominator of 7.
First, we'll simplify the two given square roots:
√7 ≈ 2.646
√8 ≈ 2.828
Next, we'll find a number between these two values. Let's consider the number 2.7 (approximately) since it is between 2.646 and 2.828.
Finally, we write the number 2.7 as an improper fraction with a denominator of 7:
2.7 = 27/10
To convert this fraction to an improper fraction, we multiply the numerator and denominator by 7:
27/10 = (27 * 7) / (10 * 7) = 189/70
Therefore, the rational number with a denominator of 7 that is between √7 and √8 is 189/70.
To find a rational number with a denominator of 7 that is between √7 and √8, we need to convert these square roots to rational numbers with the same denominator.
To do this, we can multiply both the numerator and denominator of each square root by √7 and √8 respectively.
√7 = (√7 * √7) / 1 = 7 / √7
√8 = (√8 * √8) / 1 = 8 / √8
Now, let's convert √7 and √8 to rational numbers with a denominator of 7:
√7 = (7 / √7) * ( √7 / √7)
= (7 * √7) / 7
= √7 / 1
= (√7 * 7) / 1
= 7√7
√8 = (8 / √8) * ( √8 / √8)
= (8 * √8) / 8
= √8 / 1
= (√8 * 8) / 1
= 8√8
Now we have the rational numbers 7√7 and 8√8.
To find a rational number between these two, we can find their average:
(7√7 + 8√8) / 2
Now let's simplify the numerator by multiplying and simplifying the square roots:
(7√7 + 8√8) / 2
= (7√7 + 8√8) / 2 * (√7 / √7)
= (7√7 * √7 + 8√8 * √7) / (2√7)
= (7 * 7 + 8 * √56) / (2√7)
= (49 + 8√56) / (2√7)
= (49 + 8√(7*8)) / (2√7)
= (49 + 8√7√8) / (2√7)
= (49 + 8√7√8) / (2√7)
= (49 + 8 * 2√7) / (2√7)
= (49 + 16√7) / (2√7)
= 49/ (2√7) + 16√7 / (2√7)
= 49/ (2√7) + 16
Since the denominator is 2√7, we can multiply the numerator and denominator by √7 to eliminate the square root in the denominator:
(49/ (2√7) + 16) * (√7 / √7)
= (49 * √7 + 16√7) / (√7 * 2√7)
= (49√7 + 16√7) / 2 * 7
= 65√7 / 14
Therefore, a rational number with a denominator of 7 that is between √7 and √8 is 65√7 / 14.