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# Using differentials, find the approximate value of each of the following: (a) 17/81 ^1 / 4

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

a)   $( 17/81) ^{1/4 }$

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Let $y = x^\frac{1}{4}$  and  $x = \frac{16}{81} \ and \ \Delta x = \frac{1}{81}$
$\Delta y = (x+\Delta x)^\frac{1}{4}-x^\frac{1}{4}$
$= (\frac{16}{81}+\frac{1}{81})^\frac{1}{4}-(\frac{16}{81})^\frac{1}{4}$
$(\frac{17}{81})^\frac{1}{4} = \Delta y + \frac{2}{3}$
Now, we know that $\Delta y$ is approximate equals to dy
So,
$dy = \frac{dy}{dx}.\Delta x \\ = \frac{1}{4x^\frac{3}{4}}.\frac{1}{81} \ \ \ \ \ \ \ (\because y = x^\frac{1}{4} \ and \ \Delta x = \frac{1}{81})\\ = \frac{1}{4(\frac{16}{81})^\frac{3}{4}}.\frac{1}{81} = \frac{27}{4\times 8}.\frac{1}{81} = \frac{1}{96}$
Now,
$(\frac{17}{81})^\frac{1}{4} = \Delta y + \frac{2}{3} = \frac{1}{96}+\frac{2}{3} = \frac{65}{96} = 0.677$
Hence, $(\frac{17}{81})^\frac{1}{4}$ is approximately equal to 0.677

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