Square roots and cube roots are not only a very important part of mathematics but are an important part of the syllabus for competitive exams as well. Questions on square roots and cube roots are expected as a part of the Quantitative Aptitude or Reasoning sections of the paper for exams for Bank PO, NABARD Grade A & B, SBI-PO, IBPS SO, RBI, Railways and SSC. After having studied square roots in the previous chapter, let us now turn to cubes and cube roots. Cube roots are used to solve cubic equations and to determine the dimensions of a 3-dimensional body of a given volume. The cube of any number or integer is indicated with 3 in superscript. The symbol of the cube root is 3√.
Questions from square roots and cube roots are not directly asked in the examinations. However, this topic will help in improving calculation speed and thus, will help you solve other types of questions swiftly and efficiently. Let us understand how a cube root can be found for any number.
WHAT ARE CUBES AND CUBE ROOTS?
When a factor (say x) is multiplied to itself three times or cubed, it results in an original non-integer number which is called a cube (say y). The factor that was multiplied to itself thrice is called the cube root of the resultant cube. Take a look at the example below.
Let's say we are finding the cube root of 512 or the cube of 8. This can be expressed as:
3√512= 8
Or, 83(8x8x8) = 512
In both equations, the cube is 512 and its root is 8.
∴ x= 8; y= 512
A cube can be expressed as a 3-dimensional body.
Unlike what we learned in the chapter on square roots, the cube root of a negative number is negative. This is because the multiplication is happening an odd number of times. Let us illustrate with an example.
3√-125= -5
The cube root of a cubed fraction is represented below.
3√ 827 = 23
Given below is a table of the cubes for 1 to 30 from the Decimal number system (as we learned in the chapter on number systems). Memorizing this table will prove helpful for all mathematical problems across topics.
Cube Root | Expanded Form | Cube |
---|---|---|
1 |
1x1x1 |
1 |
2 |
2x2x2 |
8 |
3 |
3x3x3 |
27 |
4 |
4x4x4 |
64 |
5 |
5x5x5 |
125 |
6 |
6x6x6 |
216 |
7 |
7x7x7 |
343 |
8 |
8x8x8 |
512 |
9 |
9x9x9 |
729 |
10 |
10x10x10 |
1000 |
11 |
11x11x11 |
1331 |
12 |
12x12x12 |
1728 |
13 |
13x13x13 |
2197 |
14 |
14x14x14 |
2744 |
15 |
15x15x15 |
3375 |
16 |
16x16x16 |
4096 |
17 |
17x17x17 |
4193 |
18 |
18x18x18 |
5832 |
19 |
19x19x19 |
6859 |
20 |
20x20x20 |
8000 |
Let us now try to understand how cube roots are derived from a given number.
Prime factorization method: Here, the factors being cubed are determined. Of all the factors bearing the same value, one is selected and multiplied with the selected factors from other factors. Take for example this problem below.
Find the cube root of 3375
Let us look at another example. Find the cube root of 1728.
Cube Root | Expanded Form | Cube |
---|---|---|
1 |
1x1x1 |
1 |
2 |
2x2x2 |
8 |
3 |
3x3x3 |
27 |
4 |
4x4x4 |
64 |
5 |
5x5x5 |
125 |
6 |
6x6x6 |
216 |
7 |
7x7x7 |
343 |
8 |
8x8x8 |
512 |
9 |
9x9x9 |
729 |
10 |
10x10x10 |
1000 |
11 |
11x11x11 |
1331 |
12 |
12x12x12 |
1728 |
13 |
13x13x13 |
2197 |
14 |
14x14x14 |
2744 |
15 |
15x15x15 |
3375 |
16 |
16x16x16 |
4096 |
17 |
17x17x17 |
4193 |
18 |
18x18x18 |
5832 |
19 |
19x19x19 |
6859 |
20 |
20x20x20 |
8000 |
The prime factorization method is used mostly when finding the cube root of perfect cubes.
Long division method: The long division method is also made use of when trying to find cube roots. Let us also illustrate this method with the help of an example.
Find the cube root of 8337
Estimation method: This method is commonly used when trying to find the cube roots of large perfect cubes. Such as below;
Find the cube of 97336
As we have already been told before, squares and cubes (and their roots) are not tested directly in competitive examinations. However, the knowledge of the same is elementary to solving other problems swiftly and efficiently. Memorize the squares of numbers at least till 30 and the cubes till 20. This will help you when you work on other arithmetic or quantitative aptitude topics.