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)
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Answers
Answered by
GPT 3.5
To find a rational number between $7-\sqrt7$ and $8-\sqrt7$, we need to find the average of these two numbers. Adding them and then dividing by $2$, we find that the average is $$\frac{(7-\sqrt{7})+(8-\sqrt{7})}{2} = \frac{15-2\sqrt{7}}{2}.$$ To express this as an improper fraction, we multiply both the numerator and the denominator by $2$ to get $$\frac{30-4\sqrt{7}}{4} = \boxed{\frac{15}{2}-\sqrt{7}}.$$
Answered by
GPT 3.5
I apologize for the mistake in my previous response. Let's try again to find a rational number between $7-\sqrt{7}$ and $8-\sqrt{7}$.
To find a rational number between two irrational numbers, we can take their average. The average of $7-\sqrt{7}$ and $8-\sqrt{7}$ is $$\frac{(7-\sqrt{7}) + (8 - \sqrt{7})}{2} = \frac{15 - 2\sqrt{7}}{2} = \frac{15}{2} - \sqrt{7}.$$
This is not yet in the form of a rational number with a denominator of $7$, so we need to multiply both the numerator and the denominator by $7$ to get rid of the square root: $$\left(\frac{15}{2} - \sqrt{7}\right) \cdot \frac{7}{7} = \frac{15 \cdot 7}{2 \cdot 7} - \sqrt{7 \cdot 7} = \frac{105}{14} - \sqrt{49} = \frac{105}{14} - 7 = \boxed{\frac{15}{2}}.$$
To find a rational number between two irrational numbers, we can take their average. The average of $7-\sqrt{7}$ and $8-\sqrt{7}$ is $$\frac{(7-\sqrt{7}) + (8 - \sqrt{7})}{2} = \frac{15 - 2\sqrt{7}}{2} = \frac{15}{2} - \sqrt{7}.$$
This is not yet in the form of a rational number with a denominator of $7$, so we need to multiply both the numerator and the denominator by $7$ to get rid of the square root: $$\left(\frac{15}{2} - \sqrt{7}\right) \cdot \frac{7}{7} = \frac{15 \cdot 7}{2 \cdot 7} - \sqrt{7 \cdot 7} = \frac{105}{14} - \sqrt{49} = \frac{105}{14} - 7 = \boxed{\frac{15}{2}}.$$
Answered by
GPT 3.5
Apologies for the confusion. Let's try again to find a rational number between $7-\sqrt{7}$ and $8-\sqrt{7}$.
To find a rational number between two given numbers, we need to take their average. The average of $7-\sqrt{7}$ and $8-\sqrt{7}$ is: \begin{align*}
\frac{(7-\sqrt{7}) + (8 - \sqrt{7})}{2} &= \frac{15-\sqrt{7}-\sqrt{7}}{2}\\
&=\frac{15-2\sqrt{7}}{2}\\
&=\frac{15}{2}-\sqrt{7}.
\end{align*}
So, the rational number with a denominator of 7 that is between $7-\sqrt{7}$ and $8-\sqrt{7}$ is: $$\boxed{\frac{15}{2}-\sqrt{7}}$$
To find a rational number between two given numbers, we need to take their average. The average of $7-\sqrt{7}$ and $8-\sqrt{7}$ is: \begin{align*}
\frac{(7-\sqrt{7}) + (8 - \sqrt{7})}{2} &= \frac{15-\sqrt{7}-\sqrt{7}}{2}\\
&=\frac{15-2\sqrt{7}}{2}\\
&=\frac{15}{2}-\sqrt{7}.
\end{align*}
So, the rational number with a denominator of 7 that is between $7-\sqrt{7}$ and $8-\sqrt{7}$ is: $$\boxed{\frac{15}{2}-\sqrt{7}}$$
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