Question

Using differential, find the approximate values of the following:
$\left(xxiii\right)$. ${33}^{1/5}$

Answer 1

$2.0125$

Let $y=x^{1/5}\Rightarrow \frac{dy}{dx}=\frac{1}{5x^{4/5}}$
And $x=32$ and $\Delta x=1$
then,
$\Delta y=(x+\Delta x{)}^{1/5}-x^{1/5}\Rightarrow \Delta y=(32+1{)}^{1/5}-(32{)}^{1/5}\Rightarrow \Delta y=(33{)}^{1/5}-2\Rightarrow (33{)}^{1/5}=2+\Delta y\dots (1)$
Now, Approximate change in value of $y$ is given by
$\Delta y\approx \left(\frac{dy}{dx}\right)\times \Delta x\Rightarrow \Delta y\approx \frac{1}{5x^{4/5}}\times \Delta x\Rightarrow \Delta y\approx \frac{1}{5(32{)}^{4/5}}\times \left(1\right)$
$\Rightarrow \Delta y\approx -\frac{1}{5\times 16}$
$\Rightarrow \Delta y\approx 0.0125$
From equation $\left(1\right)$
$(33{)}^{1/5}\approx 2+0.0125\therefore (33{)}^{1/5}\approx 2.0125$