How to Calculate Square Roots

1. Understand the Concept:
   • The square root of a number is a value that, when multiplied by itself, equals the original number. For example, the square root of 25 is 5 because 5 × 5 = 25.
2. Basic Methods for Calculation:
   • Using a Calculator:
     • Most calculators have a square root button (√). Simply input the number and press the square root button.
   • Estimation:
     • To estimate the square root, find two perfect squares between which your number lies. For example, to find the square root of 50, notice that 7² = 49 and 8² = 64. Thus, the square root of 50 is between 7 and 8.
   • Manual Calculation (for small numbers):
     • Babylonian Method (or Heron’s Method):
       • Start with an initial guess (x₀).
       • Improve the guess using the formula: x_{n+1} = (x_n + a / x_n) / 2
       • Repeat until the guess stabilizes.

Example Calculations

1. Finding the Square Root of 25

• 25 is a perfect square.
• Calculation:
  • The square root of 25 is 5, because 5 × 5 = 25.

2. Finding the Square Root of 50

• Estimation:
  • Find perfect squares around 50: 7² = 49 and 8² = 64.
  • Therefore, the square root of 50 is between 7 and 8.
• Refinement:
  • Using a calculator, the square root of 50 is approximately 7.071.

Quick Reference Table

Here’s a table with square roots of common numbers:

Why Calculate Square Roots?

• Mathematics: Essential for solving quadratic equations and various mathematical problems.
• Engineering: Used in calculations for areas, materials, and dimensions.
• Everyday Life: Useful in practical applications like determining dimensions in construction or design.