Question

Using differential, find the approximate value of the \[\left( 29 \right)^\frac{1}{3}\] ?

Answer 1

\[\text {Consider the function }y = f\left( x \right) = \left( x \right)^\frac{1}{3} . \]

\[\text{Let }: \]

\[ x = 27 \]

\[x + ∆ x = 29\]

\[\text { Then,} \]

\[ ∆ x = 2\]

\[\text { For } x = 27, \]

\[ y = \left( 27 \right)^\frac{1}{3} = 3\]

\[\text { Let }: \]

\[ dx = ∆ x = 2\]

\[\text { Now }, y = \left( x \right)^\frac{1}{3} \]

\[ \Rightarrow \frac{dy}{dx} = \frac{1}{3 \left( x \right)^\frac{2}{3}}\]

\[ \Rightarrow \left( \frac{dy}{dx} \right)_{x = 27} = \frac{1}{27}\]

\[ \therefore ∆ y = dy = \frac{dy}{dx}dx = \frac{1}{27} \times 2 = 0 . 074\]

\[ \Rightarrow ∆ y = 0 . 074\]

\[ \therefore \left( 29 \right)^\frac{1}{3} = y + ∆ y = 3 . 074\]