Assertion (A): If 3^x = $9\sqrt{3}$, then x = $\dfrac{5}{2}$.

Reason (R): If a is a real positive number and n is positive integer then $\sqrt[n]{a}$ is also written as $a^\frac{1}{n}$.

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

If a is a real positive number and n is positive integer then $\sqrt[n]{a}$ is also written as $a^\frac{1}{n}$.

This statement is a fundamental property of exponents and radicals. It defines the relationship between roots and fractional exponents. For example, $\sqrt{a} = a^\frac{1}{2}, \sqrt[3]{a} = a^\frac{1}{3}$.

∴ Reason (R) is true.

Given,

$\Rightarrow 3^x = 9 \sqrt{3}\\[1em] \Rightarrow 3^x = 3^2 . 3^\dfrac{1}{2}\\[1em] \Rightarrow 3^x = (3)^{2 + \dfrac{1}{2}}\\[1em] \Rightarrow 3^x = (3)^{\dfrac{4 + 1}{2}}\\[1em] \Rightarrow 3^x = (3)^{\dfrac{5}{2}}\\[1em] \Rightarrow x = \dfrac{5}{2}.$

∴ Assertion (A) is true.

∴ Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason (or explanation) for Assertion (A).

Hence, option 3 is the correct option.