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Cube root is defined as the number multiplied three times by itself in order to get a number which is the cubic value. The symbol of cube root is denoted as 3√ . The meaning of the word 'root' exactly is a primary source. When we study geometry, the cube is generally classified in the category of solids. In a solid cube, all the sides are equal to each other. The volume of the cube can be calculated by multiplying the side three times or side3.
Table of Content
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Cube Root Formula
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Let us consider a number ‘a’ and the cube root of this number is represented by ‘b’. Simplifying it like 3√a = b and this is possible if and only if, a = b3. The cube root of ‘a’ can be represented by:
Formula to Find Cube Root
Perfect Cube Roots
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The perfect cube root of a natural number is defined as the perfect factorization of a number. To quote an example, 216 is calculated when 6 is multiplied thrice with itself. Although, 215 or 217 are not considered as perfect cubes as there are no numbers which multiplied three times with themselves will not give either of these results. The perfect cube roots of the first 20 natural numbers are mentioned below:
| Cube Root | Natural Numbers |
|---|---|
| 1 | 1 |
| 8 | 2 |
| 27 | 3 |
| 64 | 4 |
| 125 | 5 |
| 216 | 6 |
| 343 | 7 |
| 512 | 8 |
| 729 | 9 |
| 1000 | 10 |
| 1331 | 11 |
| 1728 | 12 |
| 2197 | 13 |
| 2744 | 14 |
| 3375 | 15 |
| 4096 | 16 |
| 4913 | 17 |
| 5832 | 18 |
| 6859 | 19 |
| 8000 | 20 |
Steps to Find Cube Root
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The basic method of finding the cube root is prime factorization. The basic steps to finding a cube root are mentioned below:
- To begin with, start with the factorization of the number.
- Divide the factors one by one with the same number until the remainder is 1.
- If there is any factor that is not divisible, then it means that the number is not a perfect cube and its cube root cannot be calculated.
Mentioned below is an example to find cube root of 512:
Applications of Cube Root Formula
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The cube root formula has a few applications that are important for students to understand and strengthen their basics. Some of these applications are mentioned henceforth:
- It helps in solving complex cubic equations.
- It can be used to find the side of a solid cube if the volume of the cube is provided in the questions asked.
- In architectural terms, it is used in calculating the accurate dimension of the apartment/room/flat.
Read: Cubes and Cube roots MCQs
Cube Root of a Fraction
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The cube root of a fraction is calculated very similarly to finding the cube root of a natural number. The cube root of a fraction can be calculated by taking the cube roots of the numerator and denominator individually.
For example, the cube of the fraction (27/343) is written as [(3√27) / (3√343)] = 3/7.
Read: NCERT Solution Cube and roots
Cube Root of Negative Numbers
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The method to find the cube root of any negative number is the same as finding the cube of a whole number or integer. However, there is one step that should be taken care of every time. That is, when we factorise the number, the answer is always negative in value. Similarly, factoring a positive number has a positive value when its cube is calculated.
For example, the cube roots of -64, -27 and -125 are -4, -3 and -5 respectively.
Things to Remember
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- The cube root of a fraction is calculated by taking the cube roots of the numerator and denominator individually.
- The cube root of a negative number is always negative just as the number and the same goes for a positive number.
- The perfect cube root of a natural number is calculated by doing its prime factorization and calculating the three similar numbers.
- The formula of cube root is 3√ a, where ‘a’ is any number.
Solved Questions
Ques. Find the cube root of 12167 without factorization. (3 marks)
Ans. First we need to separate 12167 into two groups.
12167 → 12 and 167
Starting with the unit's digit, 167 is ending with 7 and the cube of a number that ends with a 3 is 7.
Therefore, 3 would be the unit digit of the cube root.
Now, to calculate the ten's digit.
At ten's digit, we have 2 and the cube of 2 is 8.
The cube of 3 is 27.
So, 8<12<27
Thus, the ten's digit of the cube root would be 2.
3√12167 = 23
Ques. Find the volume of the cube, when its surface area is given as 96cm². (2 marks)
Ans. The surface area of the cube = 6 x Side²
As it is given, the Surface area of the cube = 96cm²
6 x Side² = 96
Side² = 96 / 6
Side² = 16
Side = 4
The volume of the cube = Side ³ = (4)³ = 64cm³
Ques. Find the cube root of 15625. (2 marks)
Ans. By using the method of prime factorization,
3√(15625) = 3√(5 x 5 x 5 x 5 x 5 x 5)
Arranging two similar sets of 5³,
³√(15625) = (5 x 5) = 25.
Ques. Calculate the cube root of 46656 using the prime factorization method. (2 marks)
Ans. The cube root of 46656 would be calculated using the prime factorization method,
The prime factorization of 46656 is (2 x 2 x 2) x (2 x 2 x 2) x (3 x 3 x 3) x (3 x 3 x 3)
Now 3√5832= 3√(2³ x 2³ x 3³ x 3³) = 2 x 2 x 3 x 3 = 36.
Therefore, the cube root of 46656 is 36.
Ques. Calculate the cube root of 110592 using the prime factorization method. (2 marks)
Ans. The cube root of 110592 would be calculated using the prime factorization method,
The prime factorization of 110592 is (2 x 2 x 2) x (2 x 2 x 2) x (2 x 2 x 2) x (2 x 3 x 2 x 3 x 2 x 3)
Now 3√5832= 3√(8³ x 6³) = 8 x 6 = 48.
Therefore, the cube root of 110592 is 48.
Ques. Find the cube root of 287496. (2 marks)
Ans. By using the method of prime factorization,
3√(287497) = 3√(2 x 2 x 2 x 3 x 3 x 3 x 11 x 11)
Now taking a single value of 2 and 3 from their sets,
= 2 x 3 x 11 = 66
Thus, the cube of 287496 is 66.
Ques. Calculate the cube root of 658503 without the prime factorization method. (5 marks)
Ans. Starting with making sets of 3 digits, beginning from the right,
So dividing 658503 into two sets, 503 will be the first set and 658 is the second set.
The unit digit of the cube root of 658503 will be the unit digit of the cube root of 503.
Now 503 ends with 3, so its cube root will end with a 7.
That is 73 = 343 and 173 = 4913
Thus, it is proved that 7 is the unit digit of the cube root.
Now calculating the ten's digit of the cube root of 658503
Taking the second group 658.
We can calculate that 83 = 512 and 93 = 729
Since, 512<658<729, we will be choosing the smaller value and that would be 8.
Thus, the ten's digit of the cube root is 8
To conclude, the cube root of 658503 is 87.
Ques. Find the cube root of 5832. (2 marks)
Ans. The cube root of 5832 would be calculated using the prime factorization method,
The prime factorization of 5832 is 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x 3
Now 3√5832= 3√(2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x 3) = 2 x 3 x 3 = 18.
Therefore, the cube root of 5832 is 18.
Ques. Calculate the side of a cube, when its volume is given as 64cm. (2 marks)
Ans. The formula for finding the volume of a cube is the cube of its side.
Now when the volume of the cube is given, Volume = (side)3
64 = (side)3
3√(64) = side
3√(4 x 4 x 4) = side
Therefore, the side of the cube is 4cm.
Ques. Calculate the cube root of -0.729. (2 marks)
Ans. 3√(-0.7290), to begin with, first remove the decimal and put 1000 in the denominator,
3√-0.7290 = 3√-729/1000
=3√- (3 x 3 x 3 x 3 x 3 x 3)/10 x 10 x 10
=3√- (33 x 33) / 103
= - (3 x 3)/10 = -9/10 = 0.9
Therefore, the cube root of -0.729 is -0.9.
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