Properties of Cubes of Numbers


 
 
Concept Explanation
 

Properties of Cubes of Numbers

The cube of natural numbers have the following interesting properties:

Property 1:  Cubes of all even natural numbers are even.

Property 2:  Cubes of all odd natural numbers are odd.

Property 3: The sum of the cubes of first n natural numbers is equal to the square of their sum. That is,

                     1^{3}+2^{3}+3^{3}+.....+n^{3}=(1+2+3+....+n)^{2}

Property 4: Cubes of the numbers ending in digits 1, 4, 5, 6, and 9 are the numbers ending in the same digit. Cubes of numbers ending in digit 2 end in digit 8 and the cube of numbers ending in digit 8 ends in digit 2. The cubes of the numbers ending in digits 3 and 7 ends in digits 7 and 3 respectively.

Explanation: The cubes of the first 10 natural numbers are given in the following table.

Number x Cube x^{3} Number x Cube x^{3}
1 1 11 1331
2 8 12 1728
3 27 13 2197
4 64 14 2744
5 125 15 3375
6 216 16 4096
7 343 17 4913
8 512 18 5832
9 729 19 6059
10 1000 20 8000

We shall learn about the cubes of negative integers.

We have,

           (-1)^{3}=(-1)times(-1)times(-1)=-1

therefore ;;-1 is the cube of itself.

Similarly,

          (-2)^{3}=(-2)times(-2)times(-2)=-8

therefore ;;-8  is the cube of -2.

         (-3)^{3}=(-3)times(-3)times(-3)=-27

therefore ;;-27 is the cube of -3 and so on.

In general, if m is a positive integer, then

        (-m)^{3}=(-m)times(-m)times(-m)=-m^{3}

Thus, for any positive integer m,-m^{3} is the cube of -m.

Cube of a Rational Number

Let a=frac{m}{n} be a rational number (m, n are non-zero integers such that nneq pm 1) other than an integer, then the cube of a is defined as a^{3}=atimes atimes a

or,     left ( frac{m}{n} right )^{3}=frac{m}{n}timesfrac{m}{n}timesfrac{m}{n}=frac{m^{3}}{n^{3}}

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Sample Questions
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Question : 1

Which of the following statements is CORRECT?

Statement 1 : Cube root of 117.649 is a rational number .

Statement 2 : Cube of an odd number may or may not be odd.

Right Option : A
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Explanation
Question : 2

Mr. X  gave a problem to Mr. Y.

" Difference of two perfect cubes is 189. If the cube root of the smaller of the two number is 3, find the cube root of the larger number. "

Help Mr. Y to answer the question.

Right Option : B
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Explanation
Question : 3

The cube of a number x is nine time x , then find X , where x neq 0 and x neq  - 3.

Right Option : D
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Explanation
 
 


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