A physical quantity $$z$$ depends on four observables $$a$$, $$b$$, $$c$$ and $$d$$, as $$z = \frac{a^2 b^{2/3}}{\sqrt{cd^3}}$$. The percentage of error in the measurement of $$a$$, $$b$$, $$c$$ and $$d$$ are $$2\%$$, $$1.5\%$$, $$4\%$$ and $$2.5\%$$ respectively. The percentage of error in $$z$$ is:

We have the physical quantity $$z$$ expressed as

$$z \;=\; \frac{a^{2}\,b^{\,2/3}}{\sqrt{c}\,d^{3}}$$

(the square-root sign acts only on $$c$$, while $$d^{3}$$ is outside the root).

In error analysis we use the general rule: if a quantity can be written in the form

$$z \;=\; a^{\,m}\,b^{\,n}\,c^{\,p}\,d^{\,q},$$

then the fractional (relative) error in $$z$$ is obtained from

$$\frac{\Delta z}{z} =\; |m|\,\frac{\Delta a}{a} +\; |n|\,\frac{\Delta b}{b} +\; |p|\,\frac{\Delta c}{c} +\; |q|\,\frac{\Delta d}{d}.$$

This formula is nothing but the differential form of the logarithmic derivative $$\ln z = m\ln a + n\ln b + p\ln c + q\ln d$$ with all magnitudes taken positive because every separate error contributes additively to the maximum possible error.

Comparing the given expression with the standard form we read off the exponents directly:

$$m = 2,\qquad n = \frac23,\qquad p = -\frac12,\qquad q = -3.$$

Only the absolute values of these exponents matter for the percentage error, so we write

$$|m| = 2,\qquad |n| = \frac23,\qquad |p| = \frac12,\qquad |q| = 3.$$

The stated percentage errors in the measured quantities are

$$\frac{\Delta a}{a}\times100 = 2\%,\; \frac{\Delta b}{b}\times100 = 1.5\%,\; \frac{\Delta c}{c}\times100 = 4\%,\; \frac{\Delta d}{d}\times100 = 2.5\%.$$

Substituting these numbers into the error-propagation formula gives the percentage error in $$z$$:

$$ \frac{\Delta z}{z}\times100 = 2\,(2\%) + \left(\frac23\right)(1.5\%) + \left(\frac12\right)(4\%) + 3\,(2.5\%). $$

Now we carry out every multiplication one by one:

$$ 2\,(2\%) = 4\%, \\ \left(\frac23\right)(1.5\%) = 1\%, \\ \left(\frac12\right)(4\%) = 2\%, \\ 3\,(2.5\%) = 7.5\%. $$

Adding all these individual contributions we obtain

$$ \frac{\Delta z}{z}\times100 = 4\% + 1\% + 2\% + 7.5\% = 14.5\%. $$

So the percentage error in the calculated value of $$z$$ is $$14.5\%$$.

Hence, the correct answer is Option D.