8. Find three different irrational numbers between the rational numbers \frac{5}{7} and \frac{9}{11}



To find three different irrational numbers between the rational numbers \( \frac{5}{7} \) and \( \frac{9}{11} \), we first need to approximate these rational numbers as decimals:

\[
\frac{5}{7} \approx 0.7143, \quad \frac{9}{11} \approx 0.8182
\]

Thus, we are looking for three irrational numbers between approximately \( 0.7143 \) and \( 0.8182 \). Examples of irrational numbers are those that cannot be expressed as a ratio of two integers and have non-repeating, non-terminating decimal expansions. 

Here are three possible irrational numbers between \( 0.7143 \) and \( 0.8182 \):

1. \( \sqrt{0.75} \approx 0.8660 \), but we need a smaller number, so we try squaring decimals:
   \( \sqrt{0.72} \approx 0.8485 \), a bit too high.
   \( \sqrt{0.73} \approx 0.8544\), still high;
   hence perfect guess using manually finding all

Find three different irrational numbers between the rational numbers \( \frac{5}{7} \) and \( \frac{9}{11} \).

Learn how to find three different irrational numbers between the rational numbers 5/7 and 9/11. This problem covers key concepts in number theory, including the decimal approximation of fractions and the properties of irrational numbers. Suitable for students in Grades 9-10.

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