Cube Root
[tex] {p}^{3} = 8 \\ p = \sqrt[3]{8} \\ p = \sqrt[3]{ 2\times 2 \times 2} \\ p = 2[/tex]
Formula
[tex] {p}^{3} - 8 = 0[/tex]
Use the following formula.
[tex] {x}^{3} - {y}^{3} = (x - y)( {x}^{2} + xy + {y}^{2} )[/tex]
[tex](p - 2)( {p}^{2} + 2p + 4)[/tex]
The expression p²+2p+4 has the discriminant less than 0 ( D < 0 ).
Thus, remove the expression and leave only p-2
[tex]p - 2 = 0 \\ p = 2[/tex]
Substitution
The most obvious number that multiplies itself three times and equal 8 is 2.
Substitute p = 2
[tex] {p}^{3} = 8 \\ {2}^{3} = 8 \\ 2 \times 2 \times 2 = 8 \\ 8 = 8[/tex]
The equation is true, thus 2 is the answer.
