Using differentials- find the approximate value ofa- 178114b- 33-15

A: Given:(1781)^14 Let y=x^14 Differentiating w.r.t x, dydx=14x^14 1 ⇒dydx=14x^ 34 ⇒dydx=14(1x^34 ) We know, Δ y=(dydx )Δ x ⇒ (x+Δ x)^14 (x)^14=[14(1x^34 ) ...

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Byju's Answer

Standard IX

Mathematics

Laws of Exponents for Real Numbers

Using differe...

Question

Using differentials, find the approximate value of

$a$ . ${(\frac{17}{81})}^{\frac{1}{4}}$

$b$ . $(33)^{- \frac{1}{5}}$

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Solution

$a$ :

Given: ${(\frac{17}{81})}^{\frac{1}{4}}$
Let $y = x^{\frac{1}{4}}$

Differentiating w.r.t $x$ ,
$\frac{d y}{d x} = \frac{1}{4} x^{\frac{1}{4} - 1}$

$\Rightarrow \frac{d y}{d x} = \frac{1}{4} x^{- \frac{3}{4}}$

$\Rightarrow \frac{d y}{d x} = \frac{1}{4} ⎛ ⎜ ⎜ ⎜ ⎝ \frac{1}{x^{\frac{3}{4}}} ⎞ ⎟ ⎟ ⎟ ⎠$
We know,
$Δ y = (\frac{d y}{d x}) Δ x$

$\Rightarrow (x + Δ x)^{\frac{1}{4}} - (x)^{\frac{1}{4}} = ⎡ ⎢ ⎢ ⎢ ⎣ \frac{1}{4} ⎛ ⎜ ⎜ ⎜ ⎝ \frac{1}{x^{\frac{3}{4}}} ⎞ ⎟ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎥ ⎦ Δ x$

Substituting $x = \frac{16}{81}, Δ x = \frac{1}{81}$

$\Rightarrow {(\frac{17}{21})}^{\frac{1}{4}} - {(\frac{16}{81})}^{\frac{1}{4}} = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{1}{4} {⎛ ⎜ ⎜ ⎜ ⎝ \frac{1}{\frac{16}{81}} ⎞ ⎟ ⎟ ⎟ ⎠}^{\frac{3}{4}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (\frac{1}{81})$

$\Rightarrow {(\frac{17}{81})}^{\frac{1}{4}} - \frac{2}{3} = [\frac{1}{4} {(\frac{3}{2})}^{3}] (\frac{1}{81})$

$\Rightarrow {(\frac{17}{81})}^{\frac{1}{4}} - \frac{2}{3} = [\frac{1}{4} {(\frac{3}{2})}^{3}] (\frac{1}{81})$

$\Rightarrow {(\frac{17}{81})}^{\frac{1}{4}} - \frac{2}{3} = (\frac{27}{32}) (\frac{1}{81})$

$\Rightarrow {(\frac{17}{81})}^{\frac{1}{4}} = \frac{2}{3} + \frac{1}{96} = 0.677$

Hence, approximate value of ${(\frac{17}{81})}^{\frac{1}{4}} is 0.677$

$b$ :

Given: $(33)^{- \frac{1}{5}}$
Let $y = x^{- \frac{1}{5}}$
Differentiating w.r.t $x$ ,
$\frac{d y}{d x} = - \frac{1}{5} x^{- \frac{1}{5} - 1}$

$\Rightarrow \frac{d y}{d x} = - \frac{1}{5} x^{- \frac{6}{5}}$

$\Rightarrow \frac{d y}{d x} = - \frac{1}{5} ⎛ ⎜ ⎜ ⎜ ⎝ \frac{1}{x^{\frac{6}{5}}} ⎞ ⎟ ⎟ ⎟ ⎠$
We know,
$Δ y = (\frac{d y}{d x}) Δ x$

$\Rightarrow (x + Δ x)^{- \frac{1}{5}} - (x)^{- \frac{1}{5}} = ⎡ ⎢ ⎢ ⎢ ⎣ - \frac{1}{5} ⎛ ⎜ ⎜ ⎜ ⎝ \frac{1}{x^{\frac{6}{5}}} ⎞ ⎟ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎥ ⎦ Δ x$

Taking $x = 32, Δ x = 1$
$\Rightarrow (33)^{- \frac{1}{5}} - (32)^{- \frac{1}{5}} = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ - \frac{1}{5} ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{1}{(32)^{\frac{6}{5}}} ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (1)$

$\Rightarrow (33)^{- \frac{1}{5}}) - \frac{1}{(32)^{\frac{1}{5}}} = - \frac{1}{5} ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{1}{(32)^{\frac{6}{5}}} ⎞ ⎟ ⎟ ⎟ ⎟ ⎠$

$\Rightarrow (33)^{- \frac{1}{5}} - \frac{1}{2} = - \frac{1}{5} (\frac{1}{2^{6}})$

$\Rightarrow (33)^{- \frac{1}{5}} = \frac{1}{2} - \frac{1}{5} (\frac{1}{64}) = 0.497$

Thus, the approximate value of $(33)^{- \frac{1}{5}} is 0.497$

Suggest Corrections

Using differentials, find the approximate value of a. (1781)14 b. (33)−15

Using differentials, find the approximate value of

$a$ . ${(\frac{17}{81})}^{\frac{1}{4}}$

$b$ . $(33)^{- \frac{1}{5}}$