The Number System Unit Test
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Question
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 with a denominator of 7 between 7–√ and 8–√, we can first simplify the given expressions:
7–√ = 7 - sqrt(7)
8–√ = 8 - sqrt(7)
Now, we need to find a rational number between these two expressions. One way to do this is by finding the average of the two.
Average = (7 - sqrt(7) + 8 - sqrt(7))/2
= (15 - 2sqrt(7))/2
= (15/2) - sqrt(7)
To rewrite this as an improper fraction, we multiply both the numerator and denominator by 2:
(15/2) - sqrt(7) = (15 - 2sqrt(7))/2
= (15 * 2 - 2sqrt(7))/2
= (30 - 2sqrt(7))/2
= (30/2) - (2sqrt(7)/2)
= 15 - sqrt(7)
Therefore, the rational number with a denominator of 7 between 7–√ and 8–√ is 15 - sqrt(7).
so its 15/sqrt7
No, the correct answer is 15 - sqrt(7), not 15/sqrt(7). The question asks for the answer as an improper fraction, not as a division.
correct me if i am incorrect, but i think this is the answer to if we were looking for an irrational number between 7 and 8 instead of √ 7 and √ 8
i think maybe the formatting might have confused the bot?
To find a rational number with a denominator of 7 that is between 7–√ and 8–√, we need to find the rational number between the square roots.
First, let's simplify the expression 7–√ and 8–√:
7–√ = 7 - √7
8–√ = 8 - √7
Now, we can find the average of these two expressions to find the rational number between them:
(7–√ + 8–√) / 2 = (7 - √7 + 8 - √7) / 2 = (15 - 2√7) / 2 = 15/2 - √7
Since we need the rational number to have a denominator of 7, we can multiply the numerator and denominator by 7:
(15/2 - √7) * 7 = (105/2 - 7√7) / 2 = 105/14 - √7/2
Therefore, the rational number with a denominator of 7 that is between 7–√ and 8–√ is 105/14 - √7/2.
To find a rational number with a denominator of 7 that is between 7–√ and 8–√, we need to first simplify the given expressions. Let's start by simplifying 7–√.
The expression 7–√ can be rewritten as 7 - (√7).
Next, let's simplify 8–√. The expression 8–√ can be rewritten as 8 - (√7) since the square root of 7 is the same in both expressions.
Now, we have 7 - (√7) and 8 - (√7) as the two numbers we need to find a rational number between.
To find a rational number between these two expressions, we need to find a number that is close to both but has a denominator of 7.
Let's add the two expressions: (7 - (√7)) + (8 - (√7)).
This simplifies to 15 - 2√7.
Since we need a denominator of 7, we can multiply both the numerator and denominator by 7 to get rid of the square root:
(15 - 2√7) * 7 / 7 = (105 - 14√7) / 7
This gives us an improper fraction that meets the requirements: (105 - 14√7) / 7.