Question

The percentage error in the ${11}^{th}$ root of the number $28$ is approximately ____________ times the percentage error in $28$

• A. $\frac{1}{28}$
• B. $\frac{1}{11}$
• C. $11$
• D. $28$

Answer 1

The correct option is B $\frac{1}{11}$

Suppose $y=x^{11}$

$dy=11x^{10}dx$

Dividing by y on both sides, we have $\frac{\Delta y}{y}=\frac{11x^{10}\Delta x}{y}=\frac{11x^{10}\Delta x}{x^{11}}$

$\therefore \frac{\Delta y}{y}=\frac{11\Delta x}{x}$

Here, when $y=28$, $x$ will be the ${11}^{th}$ root of $28$.

$\therefore \frac{\Delta \left(28\right)}{28}=\frac{11\Delta \left(\sqrt[11]{1128}\right)}{\sqrt[11]{1128}}$

Hence, the percentage error in the $11$th root of $28$ would approximately be $\frac{1}{11}$ times the error in $28$