Question

The approximate value of ${(0.009)}^{1/3}$ is

• A. $0.2$
• B. $0.2083$
• C. $0.0032$
• D. $0.0083$

Answer 1

Answer: (B)

Use the relation; $f(x+\Delta x)=f(x)+\Delta x{f}^{\prime }(x);$
to determine the approximate value. Consider,
$f(x)={(x)}^{1/3}\Rightarrow f^{\prime }(x)=\frac{1}{3}{x}^{-2/3}$
Let $x=\frac{8}{1000}$ and $\Delta x=\frac{1}{1000}$
Now, $f^{\prime }(x+\Delta x)=f(x)+\Delta x{f}^{\prime }(x)$
$\Rightarrow$ ${(x+\Delta x)}^{1/3}=x^{1/3}+\frac{1}{3x^{2/3}}\times \Delta x$
$\Rightarrow$ ${\left(\frac{8}{1000}+\frac{1}{1000}\right)}^{1/3}={\left(\frac{8}{1000}\right)}^{1/3}+\frac{\frac{1}{1000}}{3{\left(\frac{8}{1000}\right)}^{1/3}}$
$\Rightarrow$ ${\left(\frac{9}{1000}\right)}^{1/3}=\frac{2}{10}+\frac{1}{1000\times 3\times \frac{2}{10}}$
$=\frac{1}{5}+\frac{1}{600}=\frac{121}{600}$
$\therefore$ ${(0.009)}^{1/3}=0.20166$