Question

Using differential, find the approximate value of the \[\left( 33 \right)^\frac{1}{5}\] ?

Answer 1

\[\text { Consider the function y } = f\left( x \right) = \left( x \right)^\frac{1}{5} . \]

\[\text { Let }: \]

\[ x = 32 \]

\[x + ∆ x = 33\]

\[\text{Then }, \]

\[ ∆ x = 1\]

\[\text { For } x = 33, \]

\[ y = \left( 32 \right)^\frac{1}{5} = 2\]

\[\text { Let }: \]

\[ dx = ∆ x = 1\]

\[\text { Now }, y = \left( x \right)^\frac{1}{5} \]

\[ \Rightarrow \frac{dy}{dx} = \frac{1}{5 \left( x \right)^\frac{4}{5}}\]

\[ \Rightarrow \left( \frac{dy}{dx} \right)_{x = 32} = \frac{1}{80}\]

\[ \therefore ∆ y = dy = \frac{dy}{dx}dx = \frac{1}{80} \times 1 = 0 . 0125\]

\[ \Rightarrow ∆ y = 0 . 0125\]

\[ \therefore \left( 33 \right)^\frac{1}{5} = y + ∆ y = 2 . 0125\]