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How to Calculate Square Roots Mentally

Jul 17, 2024

How to Calculate Square Roots Mentally

Introduction

  • Lecture by Preshtalwalker on calculating square roots of perfect squares in your head.
  • Historical context: MIT entrance exam in 1876 included a question requiring square roots without calculators.
  • Objective: Learn a method to calculate square roots mentally.

Memorize Squares 0-9

  • Squares and their last digits:
    • 1² ends in 1, 9² ends in 1
    • 2² ends in 4, 8² ends in 4
    • 3² ends in 9, 7² ends in 9
    • 4² ends in 6, 6² ends in 6
    • 0² ends in 0
    • 5² ends in 5

Method to Calculate Square Roots

Example 1: √1600

  1. Last digit is 0; corresponds to 0².
  2. Cross out hundreds and units digits (16 remains).
  3. Closest square to 16 is 4².
  4. Answer: 40.

Example 2: √4225

  1. Last digit is 5; corresponds to 5².
  2. Cross out last two digits (42 remains).
  3. Closest square to 42 is 6².
  4. Answer: 65.

Example 3: √841

  1. Last digit is 1; could be 1² or 9² (1 or 9 proposed).
  2. Cross out last two digits (8 remains).
  3. Closest square to 8 is 2².
  4. Answer Options: 21 or 29.
  5. Check: 25² (2×3=6, 625 < 841, choose 29).
  6. Answer: 29.

Example 4: √3969

  1. Last digit is 9; could be 3² or 7² (3 or 7 proposed).
  2. Cross out last two digits (39 remains).
  3. Closest square to 39 is 6².
  4. Answer Options: 63 or 67.
  5. Check: 65² (6×7=42, 4225 > 3969, choose 63).
  6. Answer: 63.

Example 5: √1156

  1. Last digit is 6; could be 4² or 6² (4 or 6 proposed).
  2. Cross out last two digits (11 remains).
  3. Closest square to 11 is 3².
  4. Answer Options: 34 or 36.
  5. Check: 35² (3×4=12, 12 > 11, choose 34).
  6. Answer: 34.

Example 6: √16384

  1. Last digit is 4; could be 2² or 8² (2 or 8 proposed).
  2. Cross out last two digits (163 remains).
  3. Closest square to 163 is 12².
  4. Answer Options: 128 or 122.
  5. Check: 125² (12×13=156, 156 < 163, choose 128).
  6. Answer: 128.

Adapting for Cube Roots

  • Cubic Last Digits: Each last digit is unique.
  • Example: Cube root of 39304.
    • Last digit is 4; corresponds to 4³.
    • Cross out last three digits (39 remains).
    • Closest cube to 39 is 3³.
    • Answer: 34.

Historical Importance of Arithmetic

  • Transition from Roman numerals to Indian numerals around the 1200s in Europe.
  • Indian numerals simplified arithmetic calculations.
  • Quote from French mathematician Laplace on the importance of Indian numerals.

Conclusion

  • Importance of arithmetic in history.
  • End note: Recognition of the YouTube community and closing remarks.

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