The "cube" of a number refers to raising that number to the power of 3. The cube of a number is represented using the exponent '³'. For instance, \(x^3\) indicates that the number 'x' is multiplied by itself three times (x × x × x).
When a number is multiplied by itself three times, the result is called its cube.
\(2^{3}\) is read as two cubed.
\(3^{3}\) is read as three cubed.
\(a \times a \times a\)=\(a^{3}\)
The value 'a' is the base, and the number 3 is called the index, exponent, or power.
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=\(6^{2}\)=\((1+2+3)^{2}\)
Therefore, the total of the cubes of the 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\((n-1) \times 3\)
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