Question

Using differential, find the approximate value of the \[\left( 255 \right)^\frac{1}{4}\] ?

Answer 1

\[\text { Consider the function y } = f\left( x \right) = \left( x \right)^\frac{1}{4} . \]

\[\text { Let }: \]

\[ x = 256\]

\[x + ∆ x = 255\]

\[\text { Then}, \]

\[ ∆ x = - 1\]

\[\text { For } x = 256, \]

\[ y = \left( 256 \right)^\frac{1}{4} = 4\]

\[\text { Let }: \]

\[ dx = ∆ x = - 1\]

\[\text { Now,} y = \left( x \right)^\frac{1}{4} \]

\[ \Rightarrow \frac{dy}{dx} = \frac{1}{4 \left( x \right)^\frac{3}{4}}\]

\[ \Rightarrow \left( \frac{dy}{dx} \right)_{x = 256} = \frac{1}{256}\]

\[ \therefore ∆ y = dy = \frac{dy}{dx}dx = \frac{1}{256} \times - 1 = \frac{- 1}{256}\]

\[ \Rightarrow ∆ y = \frac{- 1}{256} = - 0 . 003906\]

\[ \therefore \left( 255 \right)^\frac{1}{4} = y + ∆ y = 3 . 99609 \approx 3 . 9961\]