A natural number multiplied by itself three times gives a cube of that number, e.g. 1 × 1 × 1 = 1 2 × 2 × 2 = 8 3 × 3 × 3 = 27 4 × 4 × 4 = 64 The numbers 1, 8, 27, 64, … are called cube numbers or perfect cubes.
Perfect Cube: A number is a perfect cube if it can be expressed as n3 for some integer n. Prime Factor Test: In the prime factorization of a perfect cube, every prime factor appears in groups of three.
Cube Root of a Number
The cube root of a number is the side length of a cube whose volume is that number. is the inverse operation of cubing x. For example :
This means the cube root of 8 is 2.
Prime Factorization Method: Factorize the number, group identical factors in threes, and multiply one factor from each triplet to get the cube root.
Steps to Calculate Cube Root
1. Prime Factorize the Number Break the number down into its prime factors.
2. Group the Factors in Threes Arrange identical factors into sets of three.
8000 = (2 × 2 × 2) (2 × 2 × 2) (5 × 5 × 5)
3. Multiply One Factor from Each Triplet From each group of three identical primes, take one prime and multiply them together.
8000 = (2 × 2 × 2) (2 × 2 × 2) (5 × 5 × 5) Picking one prime from each triplet: 2 × 2 × 5 = 20
4. Result The product you get in Step 3 is the cube root of the original number.
Therefore, cube root of 8000 is
MULTIPLE CHOICE QUESTION
Try yourself: Which of the following statements is true about perfect cubes?
A
The cube of a negative number is always positive.
B
The cube of an even number is always odd.
C
The cube of a number ending in 2 will end in 8.
D
The cube of a number ending in 5 will end in 1.
Correct Answer: C
- The statement in Option A is false. The cube of a negative number is always negative. - The statement in Option B is false. The cube of an even number is always even. - The statement in Option C is true. If the digit in the one's place of a number is 2, then the ending digit of its cube will be 8. - The statement in Option D is false. The cube of a number ending in 5 will end in 5.
Therefore, the correct answer is Option C.
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Properties of Perfect Cubes
(a) Property 1
If the digit in the one’s place of a number is 0, 1, 4, 5, 6 or 9, then the digit in the one’s place of its cube will also be the same digit.
(b) Property 2
If the digit in the one’s place of a number is 2, the digit in the one’s place of its cube is 8, and vice-versa.
(c) Property 3
If the digit in the one’s place of a number is 3, the digit in the one’s place of its cube is 7 and vice-versa.
Examples of Properties 1, 2 & 3
(d) Property 4
Cubes of even natural numbers are even.
Examples of Property 4
(e) Property 5
Cubes of odd natural numbers are odd.
Examples of Property 5
(f) Property 6
Cubes of negative integers are negative.
Examples of Property 6
Some Interesting Patterns in Cubes
1. Adding consecutive odd numbers
Note that we start with [n * (n – 1) + 1] odd number.
2. Difference of two consecutive cubes:
23 – 13 = 1 + 2 * 1 * 3
33 – 23 = 1 + 3 * 2 * 3
43 – 33 = 1 + 4 * 3 * 3
53 – 43 = 1 + 5 * 4 * 3
3. Cubes and their prime factor Each prime factor of the number appears three times in its cube.
Facts That Matter
If we multiply a number by itself three times, the product so obtained is called the perfect cube of that number.
There are only 10 perfect cubes from 1 to 1000.
Cubes of even numbers are even and those of odd numbers are odd.
The cube of a negative number is always negative.
If the prime factors of a number cannot be made into groups of 3, it is not a perfect cube.
MULTIPLE CHOICE QUESTION
Try yourself: Which property states that the cubes of even natural numbers are even?
A
Property 1
B
Property 2
C
Property 4
D
Property 6
Correct Answer: C
- Property 1 states that the digit in the one's place of a number and its cube will be the same. - Property 2 states that if the digit in the one's place of a number is 2, the digit in the one's place of its cube is 8, and vice-versa. - Property 4 states that cubes of even natural numbers are even. - Property 6 states that cubes of negative integers are negative. - Among these properties, Property 4 specifically mentions that cubes of even natural numbers are even.
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Solved Examples
Example 1: Is 500 a perfect cube? Solution:
500 = 5 * 5 * 5 * 2 * 2 ∵ In the above prime factorisation 2 * 2 remain after grouping the prime factors in triples. ∴ 500 is not a perfect cube.
Example 2: Is 1372 a perfect cube? If not, find the smallest natural number by which 1372 must be multiplied so that the product is a perfect cube. Solution:
We have 1372 = 2 * 2 * 7 * 7 * 7 Since, the prime factor 2 does not appear in a group of triples. ∴ 1372 is not a perfect cube. Obviously, to make it a perfect cube we need one more 2 as its factor. i.e. [1372] * 2 = [2 * 2 * 7 * 7 * 7] * 2 or 2744 = 2 * 2 * 2 * 7 * 7 * 7 which is a perfect cube. Thus, the required smallest number = 2.
Example 3: Is 31944 a perfect cube? If not then by which smallest natural number should 31944 be divided so that the quotient is a perfect cube? Solution:
We have 31944 = 2 * 2 * 2 * 3 * 11 * 11 * 11 Since, the prime factors of 31944 do not appear in triples as 3 is left over. ∴ 31944 is not a perfect cube. Obviously, 31944 / 3 will be a perfect cube i.e. [31944] ÷3 = [2 * 2 * 2 * 3 * 11 * 11 * 11] /3 or 10648 = 2 * 2 * 2 * 11 * 11 * 11 ∴ 10648 is a perfect cube. Thus, the required least number = 3.
FAQs on Points to Remember - Cubes and Cube Roots - (Maths) Class 8
1. How can we calculate the cube root of a number?
Ans. To calculate the cube root of a number, you can use the prime factorization method or the estimation method. For the prime factorization method, you need to factorize the number into prime factors and then group them in sets of three. The cube root of each set of three factors is then multiplied together to find the final answer. For the estimation method, you can guess and check different values until you find the closest cube root.
2. What are some properties of perfect cubes?
Ans. Perfect cubes have the property that when they are multiplied by themselves twice, they result in the original number. In other words, a perfect cube is the cube of an integer. Some examples of perfect cubes include 1, 8, 27, 64, and 125.
3. Can you explain some interesting patterns in cubes?
Ans. One interesting pattern in cubes is that the cube of any odd number is also an odd number. Additionally, the sum of the first n odd numbers is always a perfect square, which can be represented as the square of n.
4. How can we identify if a number is a perfect cube or not?
Ans. To determine if a number is a perfect cube, you can take the cube root of the number and see if the result is an integer. If the cube root is an integer, then the number is a perfect cube. If the cube root is not an integer, then the number is not a perfect cube.
5. Why is it important to understand cubes and cube roots?
Ans. Understanding cubes and cube roots is important in various mathematical applications, such as in geometry, algebra, and even in everyday life. It helps in solving equations, finding areas and volumes of shapes, and making estimations. Additionally, cubes and cube roots are fundamental concepts that lay the foundation for more advanced mathematical concepts.
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Cube of a number 
is the inverse operation of cubing x.
