Cube Root


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Cube Root

Cube root of number is a value which when multiplied by itself thrice or three times produces the original value. For example, the cube root of 27, denoted as 3√27, is 3, because when we multiply 3 by itself three times we get 3 x 3 x 3 = 27 = 33. So, we can say, the cube root gives the value which is basically cubed. Here, 27 is said to be a perfect cube. From the word, cube root, we can understand what is the root of the cube. It means which number caused the cube present under the root. Usually, to find the cubic root of perfect cubes, we use the prime factorisation method. In a similar manner, we can learn the significance of square root here.

In three-dimensional geometry, when we learn about different solids, the cube defines an object which has all its faces or sides equal in dimensions. Also, the formula to find the volume of the cube is equal to side3. Hence, if we know the volume of the cube we can easily find the side length of it using cube root formula. This is one of the major applications of cube roots. It defines the cubic root of volume of the cube is equal to the side of it.

Cube Root Symbol

The cube root symbol is denoted by ‘3√’. In the case of square root, we have used just the root symbol such as ‘√’, which is also called a radical. Hence, symbolically we can represent the cube root of different numbers as: Cube root of 5 = 3√5 Cube root of 11 = 3√11 And so on.
Also, read:

Cube Root Formula

As we already know, the cube root gives a value which can be cubed to get the original value. Suppose, cube root of ‘a’ gives a value ‘b’, such that;
3√a = b This formula is only possible if and only if; a = b3
This formula is useful when we find the cubic root of perfect cubes.
Perfect Cubes:

Number Cubes
1 1
2 8
3 27
4 64
5 125
6 216
7 343
8 512
9 729
10 1000

Taking the cube root of the above cubes is very easy. But for imperfect cubes, we cannot evaluate the cube root this easily. Hence, below is the table for different values of cube root, which students can memorise to solve the questions based on this concept.

Cube Root 1 to 30

Number Cube Root (3√)
1 1.000
2 1.260
3 1.442
4 1.587
5 1.710
6 1.817
7 1.913
8 2.000
9 2.080
10 2.154
11 2.224
12 2.289
13 2.351
14 2.410
15 2.466
16 2.520
17 2.571
18 2.621
19 2.668
20 2.714
21 2.759
22 2.802
23 2.844
24 2.884
25 2.924
26 2.962
27 3.000
28 3.037
29 3.072
30 3.107

How to Find Cube Root?

To find the cubic root of a number easily, we can use the prime factorisation method. By evaluating the prime factors we can pair similar digits in a group of three and take them out as a single digit from the cubic root. Let us take an example of finding the cube root of 8. By factoring 8, we get; 8 = 2 x 2 x 2 8 = 23
Now, if we take the cubic root both the sides, then the cube of 2 cancels the cubic root. Hence, we get the value of 3√8 Therefore,
3√8 = 2
Also, check: How to find cube root by Prime factorisation and Estimation Method

How to Simplify Algebraic Cube Roots?

To simplify the algebraic cubic roots, the cubic radical should possess the following conditions:

  • Under the radical symbol, there should be no fractional value
  • There should be no perfect power factors under the cube root symbol
  • Under the cube root symbol, no exponent value should be greater than the index value.
  • If the fraction is appearing under the radical, the denominator of the fraction should not have any fraction.

While finding the cube root of any number, we will search for the factors which occur in the set of three. For example, the cube of 8 is 2. The factor of 8 is 2 x 2 x 2. 

Unlike square roots, cube roots should not be concerned with the negative values under the radical sign. Thus, perfect cubes can also possess negative values. It should be noted that perfect squares cannot be a negative value.

For example, a cube root of – 125 is -5.

Because 125 is a perfect cube, as 125 = 5 x 5 x 5

While dealing with the perfect cube numbers, separate and determine the perfect cube factors. Sometimes, it might be helpful if we separate the expression under the radical symbol. But be sure to write the cube root for each section. Also, check each section will possess the property of perfect cubes. 

Cube Root Questions

1. Find the cube root of 64.
Solution: To find the cube root of 64, we need to use the prime factorisation method. 64 = 2×2×2×2×2×2 64 = 4 × 4 × 4 64 = 43
Now taking the cube root on both the sides, we get;
3√64 = 3√(43)
3√64 = 4
2. What is the cube root of 1331?
Solution: Using the prime factorisation method, we get; 1331 = 11×11×11 1331 = 113
3√1331 = 11
3. Find the cube root of 216.
Solution: By prime factorisation, we know; 216 = 2×2×2×3×3×3 216 = 23×33
216 = (2×3)3 = 63
3√216 = 6
4. Find 3√343.
Solution: By prime factorisation 343 = 7x7x7 343 = 73
3√343 = 7
5. Evaluate the value of 3√1728.
Solution: Using prime factorisation method; 1728 = 2×2×2×2×2×2×3×3×3 1728 = 23×23x33
1728 = (2×2×3)3
1728 = 123
3√1728 = 12

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Frequently Asked Questions on Cube Roots

Define cube root

In Mathematics, the cube root of a number “a” is a number “b”, such that b3 = a. It means that the cube root of a number gives a value which when cubed gives the original number.

Can we find the cube root for negative numbers?

Yes, we can find the cube root of a negative number. For example, the cube root of -64 is -4.

What is the cube root of 512?

The cube root of 512 is 8 because 512 is a perfect cube. When 8 is multiplied thrice, we get 512.

What is the difference between the square root and cube root?

A cube root is a number, which when cubed gives the radicand, whereas the square root is a number which when squared gives the radicand. Also, the cube root of a negative number can be negative whereas the square root of a negative number cannot be negative.

How to find the cube root of a number?

The cube root of a number can be found using the prime factorization method or the long division method.

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