Lecture Notes: Finding Square Roots of Perfect Squares

Introduction

• Objective: Learn how to find the square root of large perfect squares, especially when the answer is an integer.
• Approach: Use patterns of perfect squares to identify potential square roots.

Patterns in Perfect Squares

• Notice units digit patterns in perfect squares:
  • Numbers ending in 1 or 9 when squared result in a unit digit of 1.
  • Numbers ending in 2 or 8 when squared result in a unit digit of 4.
  • Numbers ending in 3 or 7 when squared result in a unit digit of 9.
  • Numbers ending in 4 or 6 when squared result in a unit digit of 6.
  • Numbers ending in 5 when squared result in a unit digit of 5.
  • Numbers ending in 0 when squared result in a unit digit of 0.

Example Problems

Problem 1: Square Root of 1156

• Last Digit: Ends in 6, so square root ends in 4 or 6.
• First Two Digits: 11 is between 9 (3²) and 16 (4²), so pick lower, 3.
• Possible Answers: 34 or 36.
• Comparison: 1156 is closer to 900 than 1600, so choose 34.

Problem 2: Square Root of 2304

• Last Digit: Ends in 4, so square root ends in 2 or 8.
• First Two Digits: 2304 is between 1600 (40²) and 2500 (50²), closer to 50, so choose 48.
• Conclusion: Square root is 48.

Problem 3: Square Root of 4489

• Last Digit: Ends in 9, so square root ends in 3 or 7.
• First Two Digits: 4489 is between 3600 (60²) and 4900 (70²), closer to 70, so choose 67.
• Conclusion: Square root is 67.

Problem 4: Square Root of 12996

• Last Digit: Ends in 6, so square root ends in 4 or 6.
• First Two Digits: 12996 is between 12100 (110²) and 14400 (120²), closer to 110, so choose 114.
• Conclusion: Square root is 114.

Problem 5: Square Root of 24649

• Last Digit: Ends in 9, so square root ends in 3 or 7.
• First Two Digits: 24649 is between 22500 (150²) and 25600 (160²), closer to 160, so choose 157.
• Conclusion: Square root is 157.

Conclusion

• Key Technique: Use patterns and approximation to narrow down potential square roots.
• Application: Helps to find square roots of large perfect squares without a calculator.