Question

Using differential, find the approximate values of the following:
$\left(xxii\right)$. ${\left(\frac{17}{81}\right)}^{1/4}$

Answer 1

$0.677$

Let $y=x^{1/4}\Rightarrow \frac{dy}{dx}=\frac{1}{4x^{3/4}}$
And $x=\frac{16}{81}$ and $\Delta x=\frac{1}{81}$
then,
$\Delta y=(x+\Delta x{)}^{1/4}-x^{1/4}\Rightarrow \Delta y={(\frac{16}{81}+\frac{1}{81})}^{1/4}-{\left(\frac{16}{81}\right)}^{1/4}\Rightarrow \Delta y={\left(\frac{17}{81}\right)}^{1/4}-\frac{2}{3}\Rightarrow {\left(\frac{17}{81}\right)}^{1/4}=\frac{2}{3}+\Delta y\dots (1)$
Now, Approximate change in value of $y$ is given by
$\Delta y\approx \left(\frac{dy}{dx}\right)\times \Delta x\Rightarrow \Delta y\approx \frac{1}{4x^{3/4}}\times \Delta x\Rightarrow \Delta y\approx \frac{1}{4{\left(\frac{16}{81}\right)}^{3/4}}\times \frac{1}{81}$
$\Rightarrow \Delta y\approx \frac{1}{4\times \frac{8}{27}}\times \frac{1}{81}$
$\Rightarrow \Delta y\approx 0.01$
From equation $\left(1\right)$
${\left(\frac{17}{81}\right)}^{1/4}\approx \frac{2}{3}+0.01\Rightarrow {\left(\frac{17}{81}\right)}^{1/4}\approx 0.667+0.01\therefore {\left(\frac{17}{81}\right)}^{1/4}\approx 0.677$