Answer
0.0074074
Work Step by Step
Recall that for small $\Delta x $, $\Delta y\approx dy=f'(x)dx=f'(x)\Delta x $
Or $ f(a+\Delta x)-f(a)\approx f'(a)\Delta x $
Here, $ f(x)=\sqrt[3] x $, $ a=27$, $\Delta x=0.2$ and
$ f'(x)=\frac{1}{3}\times x^{\frac{1}{3}-1}=\frac{1}{3}x^{-\frac{2}{3}}$
$\implies f(27+0.2)-f(27)\approx f'(27)\Delta x $
That is, $\sqrt[3] {27.2}-\sqrt[3] {27}\approx \frac{1}{3}(27)^{-\frac{2}{3}}\times0.2=0.0074074$
