Eight Standard >> Cube and cube roots | Part -1

Cube of a number

The "cube" of a number refers to raising that number to the power of 3. It is denoted by the exponentiation symbol "³". For example, the cube of a number "x" is expressed as "\(x^3\)" and represents the result of multiplying "x" by itself twice (x × x × x).

If a number is multiplied by itself 3 times the product is the cube of the number.

 \(2^{3}\) is read as two cubed.
 \(3^{3}\) is read as three cubed.

\(a \times a \times a\)=\(a^{3}\)
Here 'a' is called the base and 3 is its power or index or exponent.

Cube of 2=\(2^{3}\)=\(2 \times 2 \times 2\)=8
Cube of 3=\(3^{3}\)=\(3 \times 3 \times 3\)=27

Some patterns of the cube:

A) \(1^{3}\)+\(2^{3}\)=1+8=9=\(3^{2}\)=\((1+2)^{2}\)
   \(1^{3}\)+\(2^{3}\)+\(3^{3}\)=1+8+27=36=\(6^{2}\)=\((1+2+3)^{2}\)

 So sum of cubes of first n natural numbers=
 \(1^{3}\)+\(2^{3}\)+----+\(n^{3}\)=\((1+2+3+----+n)^{2}\)
               =\(\left\{\frac{n(n+1)}{2}\right\}^{2}\)

B) 1= \(1^{3}\)
   Next odd numbers  
   3+5=8=\(2^{3}\)
   Next odd numbers  
   7+9=11=16+11=27=\(3^{3}\)

C) \(2^{3}\)-\(1^{3}\)=8-1=7=1+\(2 \times 1 \times 3\)
   \(3^{3}\)-\(2^{3}\)=27-8=19=1+\(3 \times 2 \times 3\)
   \(n^{3}\)-\((n-1)^{3}\)=1+n\((n-1) \times 3\)