Question

What is the approximate value of $\sqrt[5]{5242.999}?$

• A. $\frac{1214999}{4050}$
• B. $\frac{1115}{405}$
• C. $\frac{1214999}{405000}$
• D. $\frac{1214999}{40500}$

Answer 1

The correct option is C $\frac{1214999}{405000}$

Let $f\left(x\right)=x^{\frac{1}{5}}$
$\Rightarrow f^{\prime }\left(x\right)=\frac{1}{5}{x}^{-\frac{4}{5}}$
$f(a+h)=f\left(a\right)+h{f}^{\prime }\left(a\right)$
Let $a=243,h=-0.001$
$\therefore f(243-0.001)=(243{)}^{\frac{1}{5}}-(0.001)\frac{1}{5}{\left(243\right)}^{-\frac{4}{5}}$
$\Rightarrow f\left(242.999\right)=3-\frac{1}{1000}\times \frac{1}{5}\times \frac{1}{81}=3-\frac{1}{405000}=\frac{1214999}{405000}$