[ \( \mathrm{x}, \mathrm{A} \), Simplify. Assume all variables are positive. \( (64 t u)^{-\frac{1}{6}} \) Write your answer in the form \( A \) or \( \frac{A}{B^{\prime}} \) where \( A \)

To simplify \( (64 t u)^{-\frac{1}{6}} \), we can start by breaking it down. The expression under the exponent can be written as a power of its components: \[ (64 t u)^{-\frac{1}{6}} = 64^{-\frac{1}{6}} t^{-\frac{1}{6}} u^{-\frac{1}{6}}. \] Since \( 64 = 2^6 \), we have \( 64^{-\frac{1}{6}} = (2^6)^{-\frac{1}{6}} = 2^{-1} = \frac{1}{2}. \) Thus, the full expression simplifies to: \[ \frac{1}{2 t^{\frac{1}{6}} u^{\frac{1}{6}}}. \] So, the final simplified result is \( \frac{1}{2 t^{\frac{1}{6}} u^{\frac{1}{6}}} \). Now, sprinkle in some fun facts: Did you know that the number 64 is not just an ordinary number? It's a perfect square as \( 8^2 \) and also a perfect cube as \( 4^3 \)! So, it’s like that overachiever in the math class that just keeps impressing everyone! And let's not forget, the exponent negative sign is the gate to a whole new world of fractions! It's like flipping things upside down, which sometimes can make the math easier—just like flipping a pancake can lead to a delicious breakfast!