Square Root and Cube Root
Square root and Cube roots are one of the most important topics in Maths. To find the square root of any number we need to find a number which when multiplied twice by itself gives the original number. It is denoted by the symbol ‘√’. For example: √4 = √(2 × 2) = 2. Similarly, to find the cube root of any number we need to find a number which when multiplied three times by itself gives the original number. For example, ∛27 = ∛(3 × 3 × 3) = 3.
Basically, square explains the area of a square, which is equal to ‘side x side’. But what if we have to find the side of the square. In such a case we need to find the square root of the area of the square. In the same way, a cube explains the volume of the cube, which is equal to ‘side x side x side’. No, if we have to find the length of the edges of the cube, then we need to determine the cube root of the volume of a cube.
If subtraction is the opposite of addition and division is the reverse method of multiplication, then square root and cube root are the inverse processes of finding squares and cubes of numbers.
Square Root and Cube Root Table
Memorizing the squares and the square roots of the first few numbers are almost elementary and it can help you solve problems much faster rather than having to work it out. Following is square roots list of the first 12 numbers and Cube root list of the first 15 numbers.
Square Root List
Students can practice writing the squares and square root of numbers with the help to this table up to 50. This table will help to solve mathematical problems based on squares and square roots quickly. Try to memorize the square root for at least first 10 numbers. There will be even many concepts which will be introduced in higher classes where the fundamentals of square root will be used. Let us see the table below:
| NUMBER | SQUARE | SQUARE ROOT |
|---|---|---|
| 1 | 1 | 1.000 |
| 2 | 4 | 1.414 |
| 3 | 9 | 1.732 |
| 4 | 16 | 2.000 |
| 5 | 25 | 2.236 |
| 6 | 36 | 2.449 |
| 7 | 49 | 2.646 |
| 8 | 64 | 2.828 |
| 9 | 81 | 3.000 |
| 10 | 100 | 3.162 |
| 11 | 121 | 3.317 |
| 12 | 144 | 3.464 |
Cube Root List
In the above table, you have learned the squares and square root of the numbers. Here you will find the cubes and cube root of the numbers from 1 to 15. You can write the table up to 50 to have a good practice. Memorising first 10 numbers of cube root, will help to solve problems based on them quickly.
| Number | Cube | Cube root |
| 1 | 1 | 1.000 |
| 2 | 8 | 1.260 |
| 3 | 27 | 1.442 |
| 4 | 64 | 1.587 |
| 5 | 125 | 1.710 |
| 6 | 216 | 1.817 |
| 7 | 343 | 1.913 |
| 8 | 512 | 2.000 |
| 9 | 729 | 2.080 |
| 10 | 1000 | 2.154 |
| 11 | 1331 | 2.224 |
| 12 | 1728 | 2.289 |
| 13 | 2197 | 2.351 |
| 14 | 2744 | 2.410 |
| 15 | 3375 | 2.466 |
Square Root and Cube Root Symbol
Let’s understand each concept with an example: If we multiply 5 by itself, we get 25. Now, 5 is the square root of 25 and 25 is the square number. This is also known as a perfect square because it can be represented as the product of two equal integers (5×5). A square root is expressed by the symbol: √
Now, what if the number can’t be expressed as the product of two equal integers? For example, the square root of 48 is 6.92820… Numbers like these are termed as imperfect squares because they are not integers. (They are fractions or decimals instead.)
What happens when you take 3 equal integers and express their product? Well, you get a cube. For example, (5x5x5) = 125. Hence 125 is the cube and 5 is the cube root. The cube root symbol is represented by ∛.
The symbol is almost the same as the square root symbol but the number 3 is denoted on the radical symbol.
Finding Square Root and Cube Root
To find the square root of the number, we have to determine which number was squared to find the original number. For example, if we have to find the root of 16, then as we know, when we multiply 4 by 4, the result is 16. Hence, √16 = 4. Similarly, if we have to find the cube root of a number say 64, then it is easy to determine that the cube of 4 gives 64. So cube root of 64 is 4. But if the numbers are very large that it we are not able to find the roots, then we have to use the prime factorisation method. Let us see some examples.
Examples
Example 1: Find Square root of 256.
Solution: Given: The number is 256.
Prime factorisation of 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = (2 × 2 × 2 × 2)2 = 162
Taking roots on both the sides we get;
√256 = 16.
Hence, 16 is the answer.
Example 2: Find the cube root of 512.
Solution:
Given: The number is 256.
Prime factorisation of 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = (2 × 2 × 2)3 = 83
Taking the cube rootson both the sides we get;
∛512 = 8
Hence, 8 is the answer.