Welcome to our Math lesson on **How to Manually Calculate the Square Root of a Number?**, this is the fourth lesson of our suite of math lessons covering the topic of **Roots**, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

## How to Manually Calculate the Square Root of a Number?

Despite the fact that nowadays we are in the age of technology and almost every operation is carried out with computers or calculators, it is important to understand the traditional method of calculating the square root of a number, similarly to other operations we have explained in the first chapter of this course.

At first glance, the method used to manually calculate the square root of a number looks very similar to division by hand already explained in the first chapter. Let's explain how to find the square root step by step. In this paragraph, we are going to consider only perfect squares, i.e. numbers that have their square root a natural number. For example, let's take the number 729 for illustration (the calculator gives √729 = 27).

**Step 1:** Split the original number into pairs from right to left. The first digit may end up alone but this is OK.

**Step 2:** Take the closer (smaller) square root of the leftmost chunk (here, the closest square root of 7 is 2) and write it on a separate section, similar to the one used to write the quotient in the division.

**Step 3:** The number obtained is squared and the result is written below the original chunk considered.

**Step 4:** Then, we subtract the new number from the original chunk, like in division.

**Step 5:** Now, we write the other pair next to the remainder. Another thing that we do in this step is to double the number found at the beginning (2 × 2 = 4), as this number will be used in the next step. Since here 2 × 2 corresponds to 22 - an operation we did in the third step - we have not written this operation (2 × 2) but normally, it must be done.

**Step 6:** Now, we must find a suitable digit which in this case must be the unit digit of the two numbers in the operation 4 _ × _ = 329. In our case, the number required is 7 because 47 × 7 = 329. This number is written next to 2 in the upper right section. The complete outline of the operation is shown below, where all steps are highlighted in different colours.

### Example 7

Manually calculate the following square roots.

- √2,809
- √11,236

### Solution 7

- The complete outline of the operation is shown below, where all steps are highlighted in different colours. Thus,
√2,809 = 53

- Again, the complete outline of the operation is shown below, where all steps are highlighted in different colours. Thus,
√11,236 = 106

## More Roots Lessons and Learning Resources

Tutorial ID | Math Tutorial Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions |
---|---|---|---|---|---|

7.2 | Roots | ||||

Lesson ID | Math Lesson Title | Lesson | Video Lesson | ||

7.2.1 | The Meaning of Roots. Square and Cube Root | ||||

7.2.2 | Roots as Fractional Powers | ||||

7.2.3 | Properties of Roots | ||||

7.2.4 | How to Manually Calculate the Square Root of a Number? |

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