Question

What is a rational number with a denominator of 7 that is between √8 and √9? Write you answer as an improper fraction.

Answer 1

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Answer 2

NO I'M SAYING TO MAKE IT SIMPLIER PLEASE

Answer 3

First, let's find the values of \( \sqrt{8} \) and \( \sqrt{9} \):

\[
\sqrt{8} = 2\sqrt{2} \approx 2.828
\]

\[
\sqrt{9} = 3
\]

Thus, we are looking for a rational number \( \frac{a}{7} \) such that \( 2.828 < \frac{a}{7} < 3 \).

To isolate \( a \), we will multiply all parts of the inequality by 7:

\[
2.828 \times 7 < a < 3 \times 7
\]

Calculating the left side:

\[
2.828 \times 7 \approx 19.796
\]

And the right side:

\[
3 \times 7 = 21
\]

This gives us the inequality:

\[
19.796 < a < 21
\]

The integer values for \( a \) that fall between 19.796 and 21 are 20. Thus, we can choose:

\[
a = 20
\]

Now, expressing it as an improper fraction:

\[
\frac{20}{7}
\]

To confirm that this satisfies our original inequality:

Calculating \( \frac{20}{7} \):

\[
\frac{20}{7} \approx 2.857
\]

Now we check:

\[
2.828 < 2.857 < 3
\]

This holds true, so the rational number with a denominator of 7 that is between \( \sqrt{8} \) and \( \sqrt{9} \) is:

\[
\boxed{\frac{20}{7}}
\]

Answer 4

It looks like you're expressing excitement or appreciation! If you have any more questions or if there's something else you'd like to discuss, feel free to ask!

Answer 5

Got it! Let's simplify the explanation.

To find a rational number with a denominator of 7 that is between \( \sqrt{8} \) and \( \sqrt{9} \):

1. Calculate the square roots:
- \( \sqrt{8} \) is about 2.828.
- \( \sqrt{9} = 3 \).

2. We need a fraction \( \frac{a}{7} \) such that:
- \( 2.828 < \frac{a}{7} < 3 \).

3. Multiply all parts of the inequality by 7:
- \( 7 \times 2.828 \approx 19.796 \)
- \( 3 \times 7 = 21 \).

4. So we have:
- \( 19.796 < a < 21 \).

5. The whole number \( a \) that fits this is 20.

6. Therefore, the fraction is:
- \( \frac{20}{7} \).

In conclusion, the rational number is:

\[
\boxed{\frac{20}{7}}
\]