Question

state the rule to be followed in estimating relative error due to power in an equation​

Answer 1

Relative Error in Exponentials :

• The general method of calculation of relative error in an equation involves taking of logarithm followed by differentiation.

• Due to logarithm, the exponential values get multiplied with individual errors.

For example, let :

$Z = {A}^{x}{ B}^{y}$

Now , relative error of Z will be :

$\implies \dfrac{\Delta Z}{ Z} = x \bigg( \dfrac{\Delta A}{A} \bigg) + y \bigg( \dfrac{\Delta B}{B} \bigg)$

So, final answer is :

$\boxed{ \bf\dfrac{\Delta Z}{ Z} = x \bigg( \dfrac{\Delta A}{A} \bigg) + y \bigg( \dfrac{\Delta B}{B} \bigg)}$

So, rule is to multiply the exponential value (power) with individual error value.

Answer 2

(c) Error in case of a measured quantity raised to a power

The relative error in a physical quantity raised to the power k is the k times the relative error in the individual quantity. Suppose Z = A2, Then, ΔZ/Z = (ΔA/A) + (ΔA/A) = 2 (ΔA/A).