Find the approximate value of ${(1.999)^5}$.

Let $x = 2$

and $\Delta x = - 0.001$

Let $y = {x^5}$

On differentiating both sides w.r.t. $x$, we get

$\frac{{dy}}{{dx}} = 5{x^4}$

Now, $\Delta y = \frac{{dy}}{{dx}} \cdot \Delta x = 5{x^4} \times \Delta x$

$= 5 \times {2^4} \times [ - 0.001]$

$= - 80 \times 0.001 = - 0.080$

Therefore,${(1.999)^5} = y + \Delta y$

$= {2^5} + ( - 0.080)$
$= 32 - 0.080 = 31.920$