This lesson teaches two important methods for finding the square root of numbers: approximating for a quick estimate and using the division method for an exact answer.

1. Core Concepts (Short Notes)

• Perfect Square: A number that is the result of multiplying a whole number by itself (e.g., 1, 4, 9, 16, 25, 36...).

• Square Root (√): The value that, when multiplied by itself, gives the original number. The square root of a non-perfect square is a decimal number that does not terminate or repeat.

• First Approximation: The best estimate for a square root, accurate to one decimal place. This is found by testing which decimal (like 4.1, 4.2, 4.3...) when squared, gives a result closest to the original number.

• Division Method: A step-by-step "long division" style process that can find the exact square root of any positive number, including decimals.

2. Key Methods to Master

Method 1: Finding the First Approximation (e.g., for √17)

1. Bracket the Number: Find the perfect squares just below and just above your number.

   • 16 < 17 < 25

2. Find the Root Range: The square root will be between the roots of those perfect squares.

   • √16 < √17 < √25 => 4 < √17 < 5

3. Check Proximity: Is 17 closer to 16 or 25?

   • 17 - 16 = 1

   • 25 - 17 = 8

   • It's much closer to 16, so the answer will be closer to 4 (e.g., 4.1, 4.2...).

4. Test the Decimals: Square the decimal numbers starting from the closer end.

   • 4.1² = 16.81

   • 4.2² = 17.64

5. Find the Closest Square: Which of the results (16.81 or 17.64) is closer to 17?

   • 17 - 16.81 = 0.19

   • 17.64 - 17 = 0.64

   • 16.81 is closer. Therefore, the first approximation of √17 is 4.1.

Method 2: The Division Method (e.g., for √1764)

1. Pair the Digits: Starting from the right, group the digits in pairs. 17 64.

2. First Step: Find the largest integer whose square is less than or equal to the first pair (17).

   • It's 4 (since 4² = 16).

   • Write 4 as the divisor and as the first digit of the quotient (the answer). Subtract 16 from 17 to get 1.

3. Bring Down: Bring down the next pair (64) to make the new dividend 164.

4. New Divisor: Double the current quotient (4 x 2 = 8) and write it down, leaving a space: 8_.

5. Find the Next Digit: Find a digit (let's call it 'n') to put in the blank space and also as the next digit of the quotient, so that 8n × n is less than or equal to 164.

   • Try n=1: 81 × 1 = 81 (too small)

   • Try n=2: 82 × 2 = 164 (perfect match!)

6. Finalize: Write 2 as the next digit of the quotient. Subtract 164 from 164 to get 0.

   • The answer (quotient) is 42.

3. Tips & Tricks for Exams

• Memorize Squares: Know the squares of numbers from 1 to 15 by heart (1, 4, 9... 169, 196, 225). This makes the approximation method incredibly fast.

• Pairing is KEY for Division Method:

  • For whole numbers, pair from right to left: 5 29. The last "pair" might be a single digit.

  • For decimals, pair outwards from the decimal point: 12 . 32 10. You must add a zero if a digit is left alone after the decimal point.

• Rounding Rule: To find a square root correct to two decimal places, you must use the division method to calculate it to three decimal places and then round your final answer.

• Word Problems: If a question gives you the area of a square and asks for the length of a side, it is asking you to find the square root of the area.

4. Important Points to Remember

• The √ symbol always means the positive square root.

• The division method is a mechanical process. The best way to master it is to practice 5-10 problems until the steps become automatic.

• Don't mix the methods. Use approximation only when asked for a "first approximation" or a quick estimate. Use the division method when asked for an exact answer or an answer to multiple decimal places.