## Finding Square Root

The first thing you need to know to find the square root of a number is if the number is a perfect or imperfect square. A number with an imperfect square where the square root gives a fractional or decimal value. For example, perfect squares include 4, 16, and 64, while examples of imperfect squares are 5, 7, 19, and 23.

Once you know if the number is a perfect or imperfect square, you can figure out the method for finding the square root. Here are some ways you can find the square root of a number:

### By Prime Factorisation

It’s easy to calculate the square root of a number using the prime factorization method. The prime factorization of a number is the product of the prime numbers. You can only divide prime numbers by 1 or the number itself. Below, we list the steps you need to take when using this method to determine the square root of a given number:

- Divide the given number into its prime factors
- Place pairs of factors together in a way that each pair is equal.
- Select a factor from each of the pairs
- Determine the product of the factors obtained from the step above
- The product you receive is the square root of the number given

Here are some more examples to illustrate the steps above:

Number |
Prime Factorization |
Square Root |

4 |
2 x 2 |
√4 = 2 |

9 |
(3 x 3) |
√9 = 3 |

16 |
(2 x 2) x (2 x 2) |
√16 = 2 x 2 = 4 |

144 |
(2 x 2) x (2 x 2) x (3 x 3) |
√144 = 2 x 2 x 3 = 12 |

169 |
(2 x 2) x (2 x 2) x (3 x 3) |
√169 = 13 |

256 |
(13 x 13) |
√256 = 2 x 2 x 2 x 2 = 16 |

324 |
(2 x 2) x (3 x 3) x (3 x 3) |
√324 = 2 x 3 x 3 = 18 |

This method works well when the number given is a perfect square. Otherwise, we will need to use other methods.

### Repeated Subtraction Method

This is the simplest method. You can get the square root of a given number by subtracting consecutive odd numbers from the number given until it reaches 0. Hence, the number of times the subtraction takes place is the square root of the given number. Like the method above, this also works for perfect square numbers only. Below are some examples:

#### 1. Find the square root of 16

**Answer and Explanation**

- 16 - 1 = 15
- 15 - 3 = 12
- 12 - 5 = 7
- 7- 7 = 0

The number of times that we subtracted from 16 is four. Hence, √16 = 4

#### 2. Find the Square root of 25

**Answer and Explanation **

- 25 - 1 = 24
- 24 - 3 = 21
- 21 - 5 = 16
- 16 - 7 = 9
- 9 - 9 = 0

The number of times we subtracted from 25 is five times. Hence √25 = 5

#### 3. Find the square root of 9.

**Answer and Explanation**

9 - 1 = 8

8 - 3 = 5

5 - 5 = 0

The number of times we subtracted from 9 is three times. Hence √9 = 3

#### 4. Find the square root of 4.

**Answer and Explanation**

4 - 1 = 3

3 - 3 = 0

The number of times we subtracted from 4 is two times. Hence √4=2

#### 5. Find the square root of 64

**Answer and Explanation**

64 - 1 = 63

63 - 3 = 60

60 - 5 = 55

55 - 7 = 48

48 - 9 = 39

39 - 11 = 28

28 - 13 = 15

15 - 15 = 0

The number of times we subtracted from 64 is eight times. Hence √64 = 8

### By Long Division Method

The long division method is suitable for numbers that are not perfect square numbers. It divides large numbers into steps or parts, which breaks the number into sequences. This method can give you the exact square root of the number given. Below are the steps using 164 as the given value:

**Step 1**: Place a bar over the two digits that we have, i.e., 6 and 4

**Step 2**: Divide the left-most number by the most significant number whose square is less than or equal to the number in the left-most pair.

**Step 3**: Bring down the number under the next bar to the right of the remainder. Add the last digit of the quotient to the divisor. Then, to the right of the obtained sum, find a suitable number, which, together with the result of the sum, forms a new divisor for the new dividend carried down. The terminologies are explained below:

- The remainder is the remaining part after completing the division process
- The quotient is the answer obtained when we divide one number by another.
- A divisor is a number that divides another number.

**Step 4**: The new number in the quotient will have the same number as the divisor, which can be either less or equal to the dividend.

**Step 5**: The process continues using a decimal point and adding zeros in pairs underneath the remainder.

Hence, the quotient obtained is the square of the given number. Hence, the square root of 64 is 8. Below are some of the examples:

1. √64 = 8

2. √169 = 13

3. √144 = 12

4. √400 = 20

5. √1024 = 32

### By Estimation Method

This method is long, time-consuming, and has to deal with guesses. People who are familiar with finding square roots may find the method straightforward. By definition, the estimation method is a reasonable guess that can give the actual value. It can also be used for values that cannot give perfect squares. Below are some examples with their explanation:

1. Find the square root of 15.

**Answers and Explanation**

The closest perfect squares to 15 are 16 and 9. The square roots of both numbers are 4 and 3, respectively. Therefore, it’s a good guess to say the √15 lies between 3 and 4.

If we take our guesses further, √15 will be closer to 4 than 3 since 16 is closer to 15 than 9. Thus, the square root will be greater than 3.5. Furthermore, the square of 3.8 is 14.44, and that of 3.9 is 15.21.

So, the √15 is between 3.8 and 3.9. Another guess demonstrates that the √15 is 3.87

2. Find the square root of 17.

**Answer and Explanation**

The closest perfect squares to 17 are 16 and 25, with their square root of 4 and 5, respectively. The √17 should be around 4. Since 17 is closer to 16 than 25. Therefore, the √17 is 4.123.

3. Find the square root of 19.

**Answer and Explanation**

The closest perfect squares to 19 are 16 and 25, with their square root of 4 and 5, respectively. So, the √19 will be between 4 and 5. 19 is closer to 16 than 25. So, the answer will be less than 4.5. More guesses demonstrate that the answer is 4.358.

4. Find the square root of 29.

**Answer and Explanation**

The closest perfect squares to 29 are 25 and 36, with their square root of 5 and 6, respectively. So, the √29 will be between 5 and 6. 29 is closer to 25 than 36. Thus, the answer will be less than 5.5. If we take our guesses further, we find that the answer is 5.385.

5. Find the square root of 90.

**Answer and Explanation**

The closest perfect squares to 90 are 81 and 100, with their square root of 9 and 10, respectively. So, the √90 will be between 81 and 100. 90 is nearly between 81 and 100 but is closer to 81. Thus, the answer will be close to 9.5. If we guess further, we find that the answer is 9.48.