Simplifying the Sum of Cubes Expression

The problem asks to simplify the expression $(1^3 + 2^3 + 3^3 + \dots + 8^3)^{\frac{-5}{2}}$. We will use the sum of cubes formula and exponent properties.

Sum of Cubes Calculation

The sum of the first $n$ cubes is given by the formula $\sum_{i=1}^{n} i^3 = \left(\frac{n(n+1)}{2}\right)^2$. In this case, $n=8$.

Calculate the sum:

Sum $= \left(\frac{8(8+1)}{2}\right)^2 = \left(\frac{8 \times 9}{2}\right)^2 = \left(\frac{72}{2}\right)^2 = (36)^2$.

Exponent Rule Application

Substitute the calculated sum back into the original expression:

$ ( (36)^2 )^{\frac{-5}{2}} $

Using the exponent rule $(a^m)^n = a^{m \times n}$:

$ 36^{2 \times \frac{-5}{2}} = 36^{-5} $

Final Simplification

To simplify further and match the options, express the base $36$ as $6^2$:

$ 36^{-5} = (6^2)^{-5} $

Apply the exponent rule $(a^m)^n = a^{m \times n}$ again:

$ 6^{2 \times (-5)} = 6^{-10} $

The simplified form of the expression is $6^{-10}$.