Math Problem Statement

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

Solution

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

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

To solve this:

  • The number between 8 and 9 is squared to get the range: 82=648^2 = 64 and 92=819^2 = 81.
  • So, we need to find a square root of a number between 64 and 81.

Evaluate each option:

  1. 376.08\sqrt{37} \approx 6.08 (too small)
  2. 517.14\sqrt{51} \approx 7.14 (too small)
  3. 788.83\sqrt{78} \approx 8.83 (between 8 and 9)
  4. 738.54\sqrt{73} \approx 8.54 (between 8 and 9)

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

Answer: 78\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 37\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.

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Math Problem Analysis

Mathematical Concepts

Irrational Numbers
Square Roots
Estimation

Formulas

Square Root Estimation (√n)

Theorems

Irrational Number Properties

Suitable Grade Level

Grades 8-10