Square and Square Roots
Square of a number
In mathematics, we get the square of a number by a multiplication of the number by itself. This is also known as raising a number to the power of 2. If x is a number, its square will be equal to x raised to the power 2, i.e., x2.
For example, the square of 6 is denoted by 62, giving 36 as the square of 6. Therefore, 36 is the square of 6. This was the case when we have a positive number. Some examples of positive square numbers include 1, 4, 9, 16, 25, 36, 49, 64, … and so on. These numbers are perfect squares of 1, 2, 3, 4, 5, 6, 7, ….
Suppose your teacher asks you to find the square of a negative number. What will you do? In this case, always remember that the square of a negative number is always a positive number. For instance, (-6)2 = (-6) x (-6) = 36. Here, we cannot multiply -6 with 6 because then this would give a negative number.
Check this table below and note down the squares for numbers from -25 to 25. This will make your mathematical problems easier!
| Number | Square | Number | Square |
| -25 | 625 | 1 | 1 |
| -24 | 576 | 2 | 4 |
| -23 | 529 | 3 | 9 |
| -22 | 484 | 4 | 16 |
| -21 | 441 | 5 | 25 |
| -20 | 400 | 6 | 36 |
| -19 | 361 | 7 | 49 |
| -18 | 324 | 8 | 64 |
| -17 | 289 | 9 | 81 |
| -16 | 256 | 10 | 100 |
| -15 | 225 | 11 | 121 |
| -14 | 196 | 12 | 144 |
| -13 | 169 | 13 | 169 |
| -12 | 144 | 14 | 196 |
| -11 | 121 | 15 | 225 |
| -10 | 100 | 16 | 256 |
| -9 | 81 | 17 | 289 |
| -8 | 64 | 18 | 324 |
| -7 | 49 | 19 | 361 |
| -6 | 36 | 20 | 400 |
| -5 | 25 | 21 | 441 |
| -4 | 16 | 22 | 484 |
| -3 | 9 | 23 | 529 |
| -2 | 4 | 24 | 576 |
| -1 | 1 | 25 | 625 |
| 0 | 0 |
Square root, perfect square, and imperfect square
A square root is known as the inverse operation of squaring. With square root, we can find the number that made the square. For example, 42 = 16, where 16 is the square of 4, then 4 is the square root of 16.
If we look at the table above, all the squares of integers are called perfect squares because undoing the x2 operation gives a rational number. To calculate the square root of a perfect square, you can either use the prime factorization method.
Here’s a table of perfect squares from 1 to 100
| Perfect square | Square root |
| 1 | 1 |
| 4 | 2 |
| 9 | 3 |
| 16 | 4 |
| 25 | 5 |
| 36 | 6 |
| 49 | 7 |
| 64 | 8 |
| 81 | 9 |
| 100 | 10 |
Any square number that doesn’t give a rational number upon undoing the squaring operation is called an imperfect square. For example, 234 is an imperfect square of 15.297. To calculate the square root of an imperfect square, you can use the long division method.
Here’s a table of imperfect squares from 1 to 25:
| Imperfect squares | Square roots |
| 2 | 1.414 |
| 3 | 1.732 |
| 5 | 2.236 |
| 6 | 2.449 |
| 7 | 2.646 |
| 8 | 2.828 |
| 10 | 3.162 |
| 11 | 3.317 |
| 12 | 3.464 |
| 13 | 3.606 |
| 14 | 3.742 |
| 15 | 3.873 |
| 17 | 4.123 |
| 18 | 4.243 |
| 19 | 4.359 |
| 20 | 4.472 |
| 21 | 4.583 |
| 22 | 4.690 |
| 23 | 4.796 |
| 24 | 4.899 |
Question
- What is the value of squares of all odd numbers between 1 and 25?
The values of squares of odd numbers between 1 and 25 are as follows:
| Odd numbers | Square |
| 3 | 9 |
| 5 | 25 |
| 7 | 49 |
| 9 | 81 |
| 11 | 121 |
| 13 | 169 |
| 15 | 225 |
| 17 | 289 |
| 19 | 361 |
| 21 | 441 |
| 23 | 529 |
| 25 | 625 |
Frequently Asked Questions
1. What do you mean by the square root of numbers from 1 to 25?
Ans. When someone asks you to find the square root of any number between 1 and 25, they are basically asking you to find the number that gives that number when multiplied with itself.
2. How can you find the square root of a number?
Ans. You can find the square root of perfect squares by the prime factorization method, whereas for imperfect squares, the long division method can be used.