Question

Using differential, find the approximate value of the \[\left( 82 \right)^\frac{1}{4}\] ?

Answer 1

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

\[\text { Let }: \]

\[ x = 81 \]

\[x + ∆ x = 82\]

\[\text { Then}, \]

\[ ∆ x = 1\]

\[\text { For } x = 81, \]

\[ y = \left( 81 \right)^\frac{1}{4} = 3\]

\[\text { Let }: \]

\[ dx = ∆ x = 1\]

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

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

\[ \Rightarrow \left( \frac{dy}{dx} \right)_{x = 81} = \frac{1}{108}\]

\[ \therefore ∆ y = dy = \frac{dy}{dx}dx = \frac{1}{108} \times 1 = 0 . 009259\]

\[ \Rightarrow ∆ y = 0 . 009259\]

\[ \therefore \left( 82 \right)^\frac{1}{4} = y + ∆ y = 3 . 009259\]