If $y=x^n$, then the ratio of relative errors in $y$ and $x$ is

$y=x^{n}$

$\Rightarrow \displaystyle \dfrac{dy}{dx}=nx^{n-1}$

Approximate error in y is $\displaystyle dy=\left (\dfrac{dy}{dx}\right) \Delta x$

$=nx^{n-1} \Delta x$

Relative error in y is $\displaystyle \dfrac{dy}{y}=\dfrac{n}{x}\Delta x$

Approximate error in x is $\displaystyle dx=\left (\dfrac{dx}{dy}\right) \Delta y$

$\displaystyle=\dfrac{1}{nx^{n-1}} \Delta y$

Relative error in x is $\displaystyle \dfrac{dx}{x}=\dfrac{1}{nx^{n}}\Delta y$

Required ratio $\displaystyle = \dfrac{\dfrac{n}{x}\Delta x}{\dfrac{1}{nx^{n}}\Delta y}$

$\displaystyle =n^{2}x^{n-1} \dfrac{\Delta x}{\Delta y}$

$\displaystyle =\dfrac{n}{1}$

So, the ratio of relative errors in y and x is $n:1$.