We want to simplify
[tex](-64)^{\frac{3}{2}}[/tex]When we exponentiate a number by a fraction, we exponentiate the number by the numerator, and the denominator is the radical of the root. Since our denominator is 2, we have a square root.
[tex](-64)^{\frac{3}{2}}=\sqrt[]{(-64)^3}[/tex]Expanding this expression, we have
[tex]\begin{gathered} \sqrt[]{(-64)^3}=\sqrt[]{(-1)^3(64)^3} \\ =\sqrt[]{(-1)^{}(64)^2(64)} \\ =64\sqrt[]{(-1)(64)} \\ =64\sqrt[]{(-1)(8)^2} \\ =64\cdot8\sqrt[]{(-1)^{}} \\ =512\sqrt[]{-1^{}} \end{gathered}[/tex]The square root of minus one is also know as the imaginary number.
[tex]512\sqrt[]{-1^{}}=512i[/tex]