If $\mathrm{y}={\mathrm{x}}^{\mathrm{n}}$, then ratio of relative errors in y and x is

$\begin{array}{l}\mathrm{y}={\mathrm{x}}^{\mathrm{n}}\\ \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}={\mathrm{nx}}^{\mathrm{n}-1}\\ \text{ Approximate error in }y\text{ is }dy=\left(\frac{dy}{dx}\right)\mathrm{\Delta }x\\ =n{x}^{n-1}\mathrm{\Delta }x\\ \text{ Relative error in }\mathrm{y}\text{ is }\frac{\mathrm{dy}}{\mathrm{y}}=\frac{\mathrm{n}}{\mathrm{x}}\mathrm{\Delta x}\\ \begin{array}{c}\text{ Approximate error in }\mathrm{x}\text{ is }\mathrm{dx}=\left(\frac{\mathrm{dx}}{\mathrm{dy}}\right)\mathrm{\Delta y}\\ =\frac{1}{n{x}^{n-1}}\mathrm{\Delta y}\\ \text{ Relative error in }\mathrm{x}\text{ is }\frac{\mathrm{dx}}{\mathrm{x}}=\frac{1}{{\mathrm{nx}}^{\mathrm{n}}}\mathrm{\Delta y}\\ \begin{array}{r}\text{ Required ratio }=\frac{\frac{\mathrm{n}}{\mathrm{x}}\mathrm{\Delta x}}{\frac{1}{{\mathrm{nx}}^{\mathrm{n}}}\mathrm{\Delta y}}\\ ={\mathrm{n}}^2{\mathrm{x}}^{\mathrm{n}-1}\frac{\mathrm{\Delta x}}{\mathrm{\Delta y}}\\ =\frac{\mathrm{n}}{1}\\ \text{ So, the ratio of relative errors in }\mathrm{y}\text{ and }\mathrm{x}\text{ is }\mathrm{n}:1\text{. }\end{array}\end{array}\end{array}$