# NABARD Grade A Phase I

## Cube Roots

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

• 3√=125= -5x-5x-5
• When the first two factors are multiplied, they result in a positive number (2 negatives make a positive). However, when this positive is multiplied for a third time with a negative factor then, the result is negative because when a negative and positive factor is multiplied the result is always negative.

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 RootExpanded FormCube
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

• 3√3375=x
• 3√3375= 3x3x3x5x5x5
• 3√3375= (3x5)3
• Now remove the cube from outside the bracket and simply multiply the factors within.
• 3√3375= 3x5= 15
• 3√3375= 15
•  x = 15

Let us look at another example. Find the cube root of 1728.

• 3√1728=x
• 3√1728= 2x2x2x2x2x2x3x3x3
• 3√1728= (2x2x3)
• Now remove the cube from outside the bracket and simply multiply the factors within.
• 3√1728= 2x2x3
• 3√1728= 12
•  x = 12

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

• Begin by grouping 3 digits from the back.
• So, 337 have been grouped while the 8 in the thousandth place stands alone. Now we can begin dividing
• Since 8 is a perfect cube, we can automatically go ahead and write 2 as the first quotient.
• We need to find the second half of the quotient, let’s call it y. The quotient is 2y.
• Since there has been no remainder, the dividend left now is 337. We need to find a suitable number for y to determine what can be multiplied to it to get the nearest cube.
• Let us take the first number, i.e., 1
• 213 = 9261
• Since 9261 is way over our estimation, we will select the lower number, i.e., 0.
• Thus, the 2y= 20. But the cube of twenty is not 8000, not 8337.
• So, we will keep dividing into decimals till we derive the correct answer.
•  The cube root of 8337 is 20.27
• 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

• Start by making a group of three digits starting from the rightmost digit as we did in the long division method as well
• So, the two groups are 97 and 336.
• Since the 1st group (from right) ends with a 6, we can assume that the last digit ends with either 4 or 6. So now we have two possibilities; y4 and y6.
• To determine y, let us take the second group of 97 and find the nearest perfect cube. The two nearest cubes are 27 (33) and 125 (53).
• Since 125 is closer to 97 than 27, we can assume that the number lies between 4 and 5. Since this is a perfect cube, the factor is a real whole number. ∴We are left with 4.
• Now, y4= 44 and y6= 46.
• Find the cubes of both these numbers.
• The cube of 44 is 85184 and the cube of 46 is 97336
•  The correct answer is 46.
• 3√97336= 46.

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.