find the cube of the number
1). 16
Give the process please
Answers
Step-by-step explanation:
The cube of 16 = 16³ = 16 x 16 x 16 = (16x16) x 16 = 256 x 16 = 4096
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Answer:
If you are acquainted with the famous Numerical technique known as Newton - Raphson's method , it will be an easy job to find the cube - roots of 16 ( in fact that of any positive real number) . Let (16)^(1/3) = x then it gives x^3 = 16 or f(x) = x^3 - 16 =0. Guess the initial approximation of a real root of f(x) = 0 as x1 = 2· 5 (say) then a better approximation x2 =(x1 + h) of x1 can be obtained by Newton- Raphson's iteration process with h given by ;
h = - f(x)/f'(x) = - [(x^3 - 16)/(3x^2)] with x = x1 .
= 16 - (2·5)^3 / 3 (2 ·5)^2 = 0.02 . Therefore the next better approximation is x2 = 2.5 + 0.02 = 2.52 . Now the third better one is x3 = x2 + h , where
h = 16 - (2.52)^3 /3 (2.52)^2 = - 0.000158 so that x3 = 2.52 - 0.000158 = 2.5198 .
Similarly if we go on further to find x4 = x3 + h , we see that there is no change , infact x4 is almost equal to x3. Hence no chance of any further improvement,therefore we stop the process of improvement (iterations) and accept x = 2.5198 as the real cube roots of 16 .
Further if you are working in complex field, then we get three numbers as the cube roots of 16 and these numbers are ;
x = (16)^(1/3) = 2.5198 , (2.5198 ×w) &
(2.5198 ×w^2) where w is cube roots of unity .
Step-by-step explanation:
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