Find the approximate value of `root(3)(27.027)` .

Explanation

Let `f(x)=root(3)(x)`
`:.f'(x)=("d")/("dx")(x^((1)/(3)))=(1)/(3)x^(-(2)/(3))=(1)/(3x^((2)/(3)))`
Take a = 27 and h = 0.027.
Then `f(a)=f(27)=root(3)(27)=3`
`f'(a)=f'(27)=(1)/(3(27)^((2)/(3)))=(1)/(27)`
The formula for approximation is
`f(a+h)=f(a)+hf'(a)`
`:.root(3)(27.027)=f(27.027)=f(27+0.027)`
`=f(27)+0.027.f'(27)`
`=3+0.027xx(1)/(27)`
`=3+0.001=3.001`
`:.root(3)(27.027)=3.001`.