Using differentials, find the approximate value of each of the following :

(a) ${\left( {\cfrac{{17}}{{81}}} \right)^{1/4}}$

( b) ${(33)^{1/5}}$

(a) Let us assume :

$f(x) = {x^{1/4}} \Rightarrow f'(x) = \cfrac{1}{4}{x^{ - 3/4}}$

Let's take, $x = \cfrac{{16}}{{81}}$ and $\Delta x = \cfrac{1}{{81}}$

As, $f(x + \Delta x) \approx f(x) + \Delta xf'(x) \Rightarrow {(x + \Delta x)^{1/4}} \approx {x^{1/4}} + \cfrac{{\Delta x}}{{4{x^{3/4}}}}$

$\Rightarrow {\left( {\cfrac{{16}}{{81}} + \cfrac{1}{{81}}} \right)^{14}} \approx {\left( {\cfrac{{16}}{{81}}} \right)^{1/4}} + \cfrac{{81}}{{4{{\left( {\cfrac{{16}}{{81}}} \right)}^{3/4}}}}$

$\Rightarrow {\left( {\cfrac{{17}}{{81}}} \right)^{1/4}} \approx \cfrac{2}{3} + \cfrac{1}{{81 \times 4 \times {{\left( {\cfrac{2}{3}} \right)}^3}}} = \cfrac{2}{3} + \cfrac{1}{{96}} = \cfrac{{65}}{{96}}$
Therefore, ${\left( {\cfrac{{17}}{{81}}} \right)^{1/4}} \approx 0.677.$

(b) Let us assume :

$f(x) = {x^{ - 1/5}} \Rightarrow f'(x) = \cfrac{{ - 1}}{5}{x^{ - 6/5}}.$

Let's take $x = 32$ and $\Delta x = 1$
As, $f(x + \Delta x) \approx f(x) + \Delta xf'(x) \Rightarrow {(x + \Delta x)^{ - 1/5}} \approx {x^{ - 1/5}} - \cfrac{{\Delta x}}{{5{x^{6/5}}}}$

$\Rightarrow {(33)^{ - 1/5}} \approx {(32)^{ - 1/5}} - \cfrac{1}{{5{{(32)}^{6/5}}}} = \cfrac{1}{2} - \cfrac{1}{{5 \times {2^6}}}$.

$= 0.5 - \cfrac{1}{{320}} = 0.5 - 0.003$
Therefore,${(33)^{ - 1/5}} \approx 0.497.$