Math Problem Statement

Approximate using perfect squares. What is the square root of 78?

Solution

This worksheet asks to approximate the square root of 78 using perfect squares.

  1. First, note that 7878 is between two perfect squares: 64and81(since 64=82 and 81=92).64 \quad \text{and} \quad 81 \quad \text{(since} \ 64 = 8^2 \ \text{and} \ 81 = 9^2\text{)}. Therefore, 64<78<8164 < 78 < 81.

  2. Since 64<78<8164 < 78 < 81, the square root of 78 lies between 64\sqrt{64} and 81\sqrt{81}, which simplifies to: 8<78<9.8 < \sqrt{78} < 9.

The conclusion is: 78 is between 8 and 9.\sqrt{78} \ \text{is between} \ 8 \ \text{and} \ 9.

Would you like a more detailed explanation or additional assistance on this?

Here are 5 related questions to further explore this topic:

  1. How can you estimate 78\sqrt{78} more precisely using interpolation?
  2. What is the decimal approximation of 78\sqrt{78}?
  3. How do you determine the square roots of non-perfect squares?
  4. Why does 78\sqrt{78} fall between 8 and 9 specifically?
  5. How can you find the square root of any number using long division?

Tip: To improve your estimation of square roots, consider using decimal approximations and methods like interpolation between perfect squares.

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Math Problem Analysis

Mathematical Concepts

Number Sense
Square Roots
Estimation

Formulas

Square root approximation: if a < x < b, then √a < √x < √b

Theorems

Perfect square properties
Inequalities for square roots

Suitable Grade Level

Grades 6-8