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A cube is a solid object in geometry with all edges equal and perpendicular (at a 90o degree angle) to each other.
- Cubes are exponents (mathematical expressions or numbers expressed as an, where a is the base and n is the power or exponent to which a is raised) raised to the power 3 (x3).
- A number multiplied three times gives its cube in the cube formula.
- A two-dimensional square represents a square number.
- A three-dimensional cube can represent a cube number.
- There are various properties of cube roots that can help in computing any cube number.
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Key Takeaways: Cube Roots, Perfect Cube, Natural Numbers, 3√, Patterns, Factorization Method, Estimation Method, Triangular Numbers
Cube Numbers
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When a natural number x is expressed as y (a natural number), then x will be the cube number of y. The digits 1, 8, and 27 are cube numbers of 1, 2, and 3. Hence, a number is calculated when a number is multiplied by itself three times.
M3 can be written as M x M x M
Let us say, the sides of the given cube are M, where M=2
M3= 23= 2 x 2 x 2= 8
Perfect Cube
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4 x 4 x 4 = 64
7 x 7 x 7 = 343
The numbers mentioned above are perfect cubes. As a result, if a natural number is a cube of another natural number, it is a perfect cube. In other terms, a natural number x is a perfect cube if its cube is y, x3= y.
To find a perfect cube, divide a cube number into its prime factors using the prime factorization method. Every factor appears three times in a number, resulting in a perfect cube.
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Patterns in Cube Numbers
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Adding Consecutive Odd Numbers
1=1=13
3+5=8=23
7+9+11=27=33
In the above pattern, adding consecutive odd numbers gives the sum that is further a cube of a natural number in the number series.
Prime Factors of Cubes
63 = 216 = 2x2x2x3x3x3 = 23 x 33
123 = 1728 = 2x2x2x2x2x2x3x3x3 = 23 x 23 x 33
The prime factor of a number appears three times in the prime factorization of its cube in the pattern above.
Patterns in triangular Numbers
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While observing the sums of cubes numbers, it can be said that the square of the xth triangular number is equal to the sum of the first x cube numbers.
For example: The first cube number is 1 (13) which is also equal to 12
The sum of first two cube numbers = 13 + 23 = 1 + 8 = 9 = 32
The sum of first three cube numbers = 13 + 23 + 33 = 1 + 8 + 27 = 36 = 62
The sum of first four cube numbers = 13 + 23 + 33 + 43 = 1 + 8 + 27 + 64 = 100 = 102
And so on.
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Cube Roots
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Cube roots can be found by reversing the process of finding the cube of a number. The symbol of the cube root is represented by 3√.
As seen in the previous section, 43 = 64. Hence, the cube root of 3√64 will be 4.
Cube roots can be found with the help of the prime factorization method.
Step 1: Break the number into its prime factors. Say 5832 is the given number.
5832= 2x2x2x3x3x3x3x3x3
Step 2: Group the same prime factors into three.
2x2x2 x 3x3x3 x 3x3x3
Step 3: Multiply one prime factor from each of the groups. The required cube root is their product.
2 x 3 x 3= 18
Hence, 3√5832= 18
Cube roots of a number can also be found through the estimation method.
Step 1: Start making groups of three digits, starting with the rightmost digit of any cube number, such as 117649. As a result, the two groups will be the first (117) and the second (649).
Step 2: The unit digit of the cube root will be determined by the unit digit of the first group (649). The cube roots unit's digit is 9 since the number 649 ends in 9.
Step 3: Find the cube of numbers that the second group lies between 117 which is the other group.
As 43= 64 and 53= 125
64<117<125
As the tenth digit of the cube root, choose the lesser value between 4 and 5. As a result, the cube root of 117649 is 49.
Different Properties of Cubes
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Cubes of natural numbers have certain properties:
- All even natural number cubes are even.
- All odd natural number cubes are odd.
- The square of the sum of the cubes of the first n natural numbers is equal to their sum.
- Numbers ending in the digits 4, 5, 6, and 9 are cubes of numbers ending in the same digit. The cubes of numbers that end with the digit 2 end with the digit 8. And the cubes of numbers that end with digits 8 ends with the digit 2. The cubes of numbers ending in three digits and seven digits end in three digits and seven digits, respectively.
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Points to Remember
Following are some important points:
- The cubes of any natural number can be found using either the factorization method or the estimation method.
- A sequence of the numbers 1, 3, 6, 10, 15 so on, is known as a triangular number. It is calculated by adding the natural numbers multiple times.
- Cubes of even numbers are always even and cubes of odd numbers are always odd.
- 1729 is known as Hardy- Ramanujan number as it is the smallest number that can be stated in two ways as the sum of two cubes. There are infinite such numbers.
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Sample Questions
Ques: Is the number 243 a perfect cube? (2 Marks)
Ans: 243 = 3 x 3 x 3 x 3 x 3 = 33 x 32
After arranging the 3's in triplets, 32 remains in the above factorization. As a result, the number 243 is not a perfect cube.
Ques: Is the number 392 a perfect cube? If not, identify the smallest natural number when multiplied by 392 to produce a perfect cube. (3 Marks)
Ans: 392= 2 x 2 x 2 x 7 x 7
In a group of three, the prime factor 7 does not appear. As a result, 392 is not an ideal cube. One more 7 is required to make it a cube.
397 x 7= 2x2x2x7x7x7= 2744
2744 is a perfect cube. As a result, the smallest natural number is 7.
Ques: A girl makes a plasticine cuboid with sides of 5 cm, 2 cm, and 5 cm. To make a cube, how many of these cuboids will she need? (3 Marks)
Ans: Volume of cube= 5x2x5= 50
LCM of the volume of cube= 2x5x5
As, 2, 5, and 5 cannot be grouped into triples of equal factors, we will it a perfect cube
=2 x 2 x 2 x 5 x 5 x 5 =20 x (2x5x5)
Hence, 20 is the required number of cuboids
Ques: Observe the pattern below and mention the next three rows
13= 1
13 + 23 = (1 + 2)2
13 + 23 + 33 = (1 + 2 + 3)2 (2 Marks)
Ans: 13 + 23 + 33 + 43 = (1 + 2 + 3 + 4)2
13 + 23 + 33 + 43 + 53 = (1 + 2 + 3 + 4 + 5)2
13 + 23 + 33 + 43 + 53 + 63 = (1 + 2 + 3 + 4 + 5 + 6)2
Ques: Express 73 as the sum of odd numbers. (3 Marks)
Ans: 1=1=13
3+5=8=23
7+9+11=27=33
13+15+17+19=64=43
With the help of above pattern, 73 can be written as
43+45+47+49+51+53+55=343= 73
Ques: If a number's square ends in 5, its cube ends in 25. State True or False. (2 Marks)
Ans: This statement is false as it is not necessary that if a number’s square ends with 5, its cube will end with 25.
Ques: Is it possible for a perfect cube to end with two zeros? (2 Marks)
Ans: No, it is not possible. A perfect cube cannot have two zeros at the end; instead, it must have three zeros.
Ques: Using the prime factorization method, find the cube root of 13824. (2 Marks)
Ans: 13824= 2x2x2 x 2x2x2 x 2x2x2 x 3x3x3= 23 x 23 x 33
Hence, 3√13824= 2x2x2x3= 24
Ques: Using the estimate approach, find the cube root of 17576. (4 Marks)
Ans: Step 1: Form three-digit groups beginning with 17576's rightmost digit.
(17, 576) is a number that has been assigned to you. In this situation, 576 has three digits and 17, has only two digits.
Step 2: In 576, the digit- 6 took its rightful place.
Use the number 6 in the one's place of the required cube root.
Step 3: Take the other group, 17.
The cube of 2 equals 8, and the cube of 3 equals 27. 17 is a number that falls between 8 and 27.
2 is the smaller number between 2 and 3
The one’s place of 2 will be 2 itself. Take 2 as the tenth place of 17576's cube root.
Hence, 3√17576= 26
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