DICTIONARY Square root

The square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical terms, the square root of a number x is denoted as √x or x^(1/2). For example, the square root of 9 is 3 because 3 * 3 = 9.

In this tutorial, we will cover the concept of square roots, different methods to find the square root of a number, examples, and exam-style questions with answers. Methods to Find the Square Root

1. Prime Factorization

To find the square root of a number using prime factorization, follow these steps:

• Find the prime factors of the given number.
• Pair the prime factors in groups of two identical factors.
• Multiply one factor from each pair to find the square root.

Example:

Find the square root of 36.

• Prime factors of 36: 2 * 2 * 3 * 3
• Pair the prime factors: (2 * 2) * (3 * 3)
• Multiply one factor from each pair: 2 * 3 = 6

So, the square root of 36 is 6.

2. Long Division Method

The long division method is a technique used to find the square root of a number with decimal places. Example:

Find the square root of 50.

• Pair the digits from the right: (50)
• Find the largest number whose square is less than or equal to 50: 7 (7 * 7 = 49)
• Subtract the result from 50: 50 - 49 = 1
• Bring down the next pair of digits (if any) and repeat the process.

So, the square root of 50 is approximately 7.

Finding the square root of 50 using the long division method:

• Step 1: Write the number 50 and separate the digits into pairs starting from the right. Since there are no decimal places, we only have one pair (50).

50

• Step 2: Find the largest number whose square is less than or equal to the first pair (50). In this case, the largest number is 7, as 7 * 7 = 49, which is less than or equal to 50. Write 7 above the pair and write 49 below the pair.

  7
-------
| 50
  49
-------

• Step 3: Subtract 49 from 50 and write the remainder below.

  7
-------
| 50
  49
-------
   1

• Step 4: Since there are no more digits to bring down, we can proceed to calculate the decimal places. Add a decimal point to the quotient (7), and add a pair of zeros to the remainder. Bring down the pair of zeros.

  7.
-------
| 50.00
  49
-------
   1 00

• Step 5: Double the quotient (7) and write it to the left of the remainder. Treat this doubled value as a single entity (in this case, 14). Find a digit (x) that, when combined with the doubled value (14), creates a number (14x) that can be multiplied by x to get a product less than or equal to the remainder (100). In this case, x = 1, as 141 * 1 = 141, which is less than or equal to 100. Write the 1 in the quotient after the decimal point and write 141 below the 100.

  7.1
-------
| 50.00
  49
-------
   1 00
    141
-------

• Step 6: Subtract 141 from 100 and write the remainder below.

  7.1
-------
| 50.00
  49
-------
   1 00
    141
-------
     59

• Step 7: Bring down the next pair of zeros (if necessary) and continue the process to find more decimal places. In this example, we will stop at one decimal place.

So, the square root of 50 is approximately 7.1 using the long division method. You can continue this process to find more decimal places if needed.

Finding the square root of 33 using the long division method:

• Step 1: Write the number 33 and separate the digits into pairs starting from the right. Since there are no decimal places, we only have one pair (33).

33

• Step 2: Find the largest number whose square is less than or equal to the first pair (33). In this case, the largest number is 5, as 5 * 5 = 25, which is less than or equal to 33. Write 5 above the pair and write 25 below the pair.

  5
-------
| 33
  25
-------

• Step 3: Subtract 25 from 33 and write the remainder below.

  5
-------
| 33
  25
-------
   8

• Step 4: Since there are no more digits to bring down, we can proceed to calculate the decimal places. Add a decimal point to the quotient (5), and add a pair of zeros to the remainder. Bring down the pair of zeros.

  5.
-------
| 33.00
  25
-------
   8 00

• Step 5: Double the quotient (5) and write it to the left of the remainder. Treat this doubled value as a single entity (in this case, 10). Find a digit (x) that, when combined with the doubled value (10), creates a number (10x) that can be multiplied by x to get a product less than or equal to the remainder (800). In this case, x = 7, as 107 * 7 = 749, which is less than or equal to 800. Write the 7 in the quotient after the decimal point and write 749 below the 800.

  5.7
-------
| 33.00
  25
-------
   8 00
     749
-------

• Step 6: Subtract 749 from 800 and write the remainder below.

  5.7
-------
| 33.00
  25
-------
   8 00
     749
-------
      51

• Step 7: Bring down the next pair of zeros and continue the process to find more decimal places. In this example, we will find 2 more decimal places.

  5.7
-------
| 33.00
  25
-------
   8 00
     749
-------
      51 00

• Step 8: Double the quotient without the decimal (57) and write it to the left of the remainder. Treat this doubled value as a single entity (in this case, 114). Find a digit (x) that, when combined with the doubled value (114), creates a number (114x) that can be multiplied by x to get a product less than or equal to the remainder (5100). In this case, x = 4, as 1144 * 4 = 4576, which is less than or equal to 5100. Write the 4 in the quotient after the 7 and write 4576 below the 5100.

  5.74
-------
| 33.00
  25
-------
   8 00
     749
-------
      51 00
        4576
-------

• 9: Subtract 4576 from 5100 and write the remainder below.

  5.74
-------
| 33.00
  25
-------
   8 00
     749
-------
      51 00
        4576
-------
         524

• Step 10: Bring down the next pair of zeros and continue the process to find more decimal places. In this example, we will find 1 more decimal place.

  5.74
-------
| 33.00
  25
-------
   8 00
     749
-------
      51 00
        4576
-------
         524 00

• Step 11: Double the quotient without the decimal (574) and write it to the left of the remainder. Treat this doubled value as a single entity (in this case, 1148). Find a digit (x) that, when combined with the doubled value (1148), creates a number (1148x) that can be multiplied by x to get a product less than or equal to the remainder (52400). In this case, x = 4, as 11484 * 4 = 45936, which is less than or equal to 52400. Write the 4 in the quotient after the second 4 and write 45936 below the 52400.

  5.744
-------
| 33.00
  25
-------
   8 00
     749
-------
      51 00
        4576
-------
         524 00
           45936
-------

• Step 12: Subtract 45936 from 52400 and write the remainder below.

  5.744
-------
| 33.00
  25
-------
   8 00
     749
-------
      51 00
        4576
-------
         524 00
           45936
-------
            6464

Now we have calculated the square root of 33 with three decimal places (5.744).