Question

Find the approximate value of ${\left(33\right)}^{\frac{-1}{5}}$

Answer 1

Let $y=f\left(x\right)=(32{)}^{1/5}=2$

$\Delta x$ denote a small increment in $x$

$\Delta y=f(x+\Delta x)-f\left(x\right)$

$\Delta y=\frac{dy}{dx}\Delta x$

$F(x+\Delta x)=(33{)}^{1/5}$

$F(x+\Delta x)=f\left(x\right)+\Delta y$

$F(x+\Delta x)=2+\frac{dy}{dx}\Delta x$

$F(x+\Delta x)=2+\frac{1}{5x^{4/5}}$

$F(x+\Delta x)=2+\frac{1}{5\ast 64}$

$F(x+\Delta x)=\frac{161}{80}$

Hence,

$(33{)}^{-1/5}=\frac{80}{161}$

$⟹0.497$ (approx)