Question
What is a rational number with a denominator of 7 that is between 7–√ and 8–√ . Write your answer as an improper fraction.
Answers
Answer
What did you just type to me? May you answer that question again?
Answered by
GPT 3.5
To find a rational number between $7-\sqrt{7}$ and $8-\sqrt{7}$, we can find the average of the two numbers. The average is \[
\frac{(7-\sqrt{7})+(8-\sqrt{7})}{2}=\frac{15-2\sqrt{7}}{2}=\frac{15}{2}-\frac{\sqrt{7}}{2}.
\]To write this as an improper fraction, we multiply the numerator and the denominator by 2: \[
\frac{15}{2}-\frac{\sqrt{7}}{2}=\frac{15\times 2}{2\times 2}-\frac{\sqrt{7}}{2}=\boxed{\frac{15}{2}-\frac{\sqrt{7}}{2}}.
\]
\frac{(7-\sqrt{7})+(8-\sqrt{7})}{2}=\frac{15-2\sqrt{7}}{2}=\frac{15}{2}-\frac{\sqrt{7}}{2}.
\]To write this as an improper fraction, we multiply the numerator and the denominator by 2: \[
\frac{15}{2}-\frac{\sqrt{7}}{2}=\frac{15\times 2}{2\times 2}-\frac{\sqrt{7}}{2}=\boxed{\frac{15}{2}-\frac{\sqrt{7}}{2}}.
\]
Answered by
GPT 3.5
Apologies for the confusion. To find a rational number between $7-\sqrt{7}$ and $8-\sqrt{7}$ with a denominator of 7, we can choose the average of the two numbers. The average is \[
\frac{(7-\sqrt{7})+(8-\sqrt{7})}{2}=\frac{15-2\sqrt{7}}{2}=\frac{15}{2}-\sqrt{7}.
\]To write this as an improper fraction with a denominator of 7, we multiply the numerator and the denominator by 7: \[
\frac{15}{2}-\sqrt{7}=\frac{15\times 7}{2\times 7}-\frac{\sqrt{7}}{1}\cdot\frac{7}{7}=\boxed{\frac{105-7\sqrt{7}}{2}}.
\]
\frac{(7-\sqrt{7})+(8-\sqrt{7})}{2}=\frac{15-2\sqrt{7}}{2}=\frac{15}{2}-\sqrt{7}.
\]To write this as an improper fraction with a denominator of 7, we multiply the numerator and the denominator by 7: \[
\frac{15}{2}-\sqrt{7}=\frac{15\times 7}{2\times 7}-\frac{\sqrt{7}}{1}\cdot\frac{7}{7}=\boxed{\frac{105-7\sqrt{7}}{2}}.
\]