Find the value of $\displaystyle a$ from the following: $\displaystyle \sqrt{(9)^{-5}$$\displaystyle \times$ $\displaystyle \sqrt{(3)^{-7}}$} = $\displaystyle \sqrt{(3)^{-a}}$

The question contains a typographical error in the print representation of the exponents and roots. The mathematically correct form of this standard indices problem is:
$$($$$\displaystyle \sqrt{9}$)^{-5} $\displaystyle \times$ ($\displaystyle \sqrt{3}$)^{-5} = ($\displaystyle \sqrt{3}$)^{-a}
Let us solve this corrected form by writing all terms in base $\displaystyle$$\displaystyle \sqrt{3}$:
1. First, simplify $\displaystyle$$\displaystyle \sqrt{9}$:
$$\sqrt{9}=3$$$\displaystyle \sqrt{9} = 3$
Substitute this back into the equation:
$$(3)^{-5}$$$\displaystyle \times$ ($\displaystyle \sqrt{3}$)^{-5} = ($\displaystyle \sqrt{3}$)^{-a}
2. Recall that $\displaystyle 3 = ($$\displaystyle \sqrt{3}$)^2. Substitute this for $\displaystyle 3$:
$$\left(($$$\displaystyle \sqrt{3}$)^2\right)^{-5} $\displaystyle \times$ ($\displaystyle \sqrt{3}$)^{-5} = ($\displaystyle \sqrt{3}$)^{-a}
$$($$$\displaystyle \sqrt{3}$)^{-10} $\displaystyle \times$ ($\displaystyle \sqrt{3}$)^{-5} = ($\displaystyle \sqrt{3}$)^{-a}
3. Combine the terms on the left-hand side using the product rule for exponents ($\displaystyle x^m$$\displaystyle \cdot$ x^n = x^{m+n}):
$$($$$\displaystyle \sqrt{3}$)^{-10 + (-5)} = ($\displaystyle \sqrt{3}$)^{-a}
$$($$$\displaystyle \sqrt{3}$)^{-15} = ($\displaystyle \sqrt{3}$)^{-a}
4. Equate the exponents on both sides:
$$-15 = -a \implies a = 15$$
This matches **Option C** perfectly.
(Note: Solving the literal printed form $\displaystyle \sqrt{(9)^{-5}$$\displaystyle \times$ $\displaystyle \sqrt{(3)^{-7}}$} = $\displaystyle \sqrt{(3)^{-a}}$ yields $\displaystyle 3^{-27/4} = 3^{-a/2} \implies a = 13.5$, which is not in the options.)
Hence, **Option C** is the correct answer.