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The question asks which irrational square root is between 8 and 9. The options given are:

1. \( \sqrt{37} \)
2. \( \sqrt{51} \)
3. \( \sqrt{78} \)
4. \( \sqrt{73} \)

### To solve this:
- The number between 8 and 9 is squared to get the range: \( 8^2 = 64 \) and \( 9^2 = 81 \).
- So, we need to find a square root of a number between 64 and 81.

### Evaluate each option:
1. \( \sqrt{37} \approx 6.08 \) (too small)
2. \( \sqrt{51} \approx 7.14 \) (too small)
3. \( \sqrt{78} \approx 8.83 \) (between 8 and 9)
4. \( \sqrt{73} \approx 8.54 \) (between 8 and 9)

Thus, both \( \sqrt{78} \) and \( \sqrt{73} \) are between 8 and 9. However, based on the question's intent, \( \sqrt{78} \) is a better fit since it's closer to 9.

**Answer: \( \sqrt{78} \)**

Do you want more details or have any questions?

### Related Questions:
1. What defines an irrational number?
2. How can you approximate square roots manually?
3. Why is \( \sqrt{37} \) smaller than 8?
4. Can a square root be both rational and irrational?
5. How do square roots relate to exponents?

**Tip:** When solving for square roots, estimate the value by squaring nearby whole numbers to establish a range.

Which irrational square root is between 8 and 9? (Options: √37, √51, √78, √73)

Find out which irrational square root is between 8 and 9 by estimating the square roots of numbers like 37, 51, 78, and 73. This problem involves understanding irrational numbers and square root estimation, suitable for students in Grades 8-10.

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