What is a rational number with a denominator of 7 that is between √7 and √8. write your answer as an improper fraction
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To find a rational number between √7 and √8, we can find the rational number that is halfway between these two values.
√7 ≈ 2.646
√8 ≈ 2.828
The rational number halfway between √7 and √8 can be found by calculating their average:
(2.646 + 2.828) / 2 = 2.737
To write this as an improper fraction, we can multiply it by 7 to get a denominator of 7:
2.737 * 7 = 19.159
Therefore, the rational number with a denominator of 7 between √7 and √8 is approximately 19.159/7.
To find a rational number with a denominator of 7 that is between √7 and √8, we can start by rationalizing the denominators of both square roots.
√7 can be rationalized by multiplying the numerator and denominator by √7:
√7 = (√7 * √7) / √7 = 7 / √7
Similarly, √8 can be rationalized by multiplying the numerator and denominator by √8:
√8 = (√8 * √8) / √8 = 8 / √8
Now we have:
√7 = 7 / √7
√8 = 8 / √8
To find a rational number between these two square roots, we can find the average of the two rationalized forms:
(7 / √7 + 8 / √8) / 2
Multiplying the numerators and denominators:
(7 * √7 + 8 * √7) / (2 * √7)
Simplifying the numerator:
(7 + 8) / √7
Combining the terms in the numerator:
15 / √7
To rationalize the denominator further, we multiply the numerator and denominator by √7:
(15 * √7) / (√7 * √7) = (15 * √7) / 7 = 15√7 / 7
Therefore, the rational number with a denominator of 7 between √7 and √8 is 15√7 / 7.
To find a rational number with a denominator of 7 between √7 and √8, we need to identify which values, when divided by 7, fall between the square roots of 7 and 8.
Let's start by dividing both sides of the equation by 7:
√7 ÷ 7 < x < √8 ÷ 7
Simplifying, we get:
√7/7 < x < √8/7
Now, let's calculate these values:
√7/7 ≈ 0.377
√8/7 ≈ 0.474
So, the rational number x that satisfies the condition is between 0.377 and 0.474.
To express this as an improper fraction, we need to convert the decimal into a fraction.
Since x is between the two values, we can approximate it to 0.4.
To convert 0.4 to a fraction, we write it as 4/10. Simplifying the fraction, we get 2/5.
Therefore, the rational number with a denominator of 7 that is between √7 and √8 is 2/5.