Suppose a number is raised to power 3. Then, we call it a cube of that number. In other words, if a number multiplies itself 3 times, then we call it a cube of that number.
Example: 2^{3} = 2 × 2 × 2 = 8
⇒ The cube of 2 is 8 .
The cube of a natural number is that natural number raised to power 3.
Thus, if n is a natural number, then, n^{3} is the cube of n , and we have n^{3} = n × n × n
A number is a perfect cube if it is a cube of some natural number.
Example: 27 is a perfect cube because there is a natural number 3 such that 27 = 3 × 3 × 3. But, 12 is not a perfect cube because there is no natural number whose cube is 12.
There are only 10 perfect cubes from 1 to 1000 . They are listed below.
This method is based upon an old Indian practice of multiplying two numbers. It is convenient for finding squares of twodigit numbers only. As the number of digits increases, this method becomes difficult and timeconsuming. So, we shall discuss it for twodigit numbers only.
This method uses the identity (a+b)^{3 }= a^{3}+ b^{3 }+ 3a^{2}b + 3ab^{2} for finding the cube of the twodigit number ab where a is the tens digit and b is the units digit, be a twodigit natural number.
Example: Find a cube of 42 by using the column method.
Sol: Let a = 4 and b = 2
Therefore, 42^{3} = 74088
The cube of a negative integer is always negative. In general, if m is a positive integer, then:
(−m)^{3}= −m × −m × −m
Example: (−2)^{3 }= −2 × −2 × −2 = −8
Let a = m/n be a rational number, where m and n are nonzero integers, then a cube of a is defined as a^{3} = a × a × a or
A number m is the cube root of a number n if n = m^{3}. In other words, the cube root of a number n is that number m whose cube gives n.
The cube root of a number n is denoted by ∛n. ∛n is also called a radical, n is called the radicand, and 3 is called the radical index.
The below table provides all the cube roots upto 1000.
The cube root formula is used to find the cube root of any number stated in radical form with the sign ∛3.
A natural number m is the cube root of a natural number n if n = m^{3} and we write it as ∛n = m
The units digit of a cube of a natural number depends upon the unit digit of the given number. The following are the rules to be followed, they are;
Example: Let us find the cube root of 64 by the method mentioned above.
Sol: In the given number 64 , the unit’s place is 4. Therefore, the digit at the one’s place in the cube root is 4 . Since no number is left after striking out the units and ten’s digit of the number. Therefore, the required cube root is 4 . i.e. ∛64 = 4
We use the below steps to find the cube root of a perfect cube by factors.
For any two integers a and b , we have
If x and a are two rational numbers such that x^{3} = a , then we say that x is the cube root of a and we write it as ∛a =x.
In other words, for any rational number a/b, we have
56 videos104 docs95 tests

1. What are cubes? 
2. What are cubes of natural numbers? 
3. What are perfect cube numbers? 
4. How can we find the cube of a twodigit number using the column method? 
5. What is the relationship between squares, square roots, cubes, and cube roots? 
56 videos104 docs95 tests


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