Simplify \( \left(\frac{81}{256}\right)^{\frac{1}{8}}\left(\frac{81}{256}\right)^{\frac{5}{8}} \). Write your answer as a fraction in lowest terms.

To simplify \( \left(\frac{81}{256}\right)^{\frac{1}{8}}\left(\frac{81}{256}\right)^{\frac{5}{8}} \), we can use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \). We start with: \[ \left(\frac{81}{256}\right)^{\frac{1}{8}} \cdot \left(\frac{81}{256}\right)^{\frac{5}{8}} = \left(\frac{81}{256}\right)^{\frac{1}{8} + \frac{5}{8}} = \left(\frac{81}{256}\right)^{\frac{6}{8}} = \left(\frac{81}{256}\right)^{\frac{3}{4}}. \] Next, we can express \( \left(\frac{81}{256}\right)^{\frac{3}{4}} \) as: \[ \left(\frac{81}{256}\right)^{\frac{3}{4}} = \frac{81^{\frac{3}{4}}}{256^{\frac{3}{4}}}. \] Now, we need to calculate \( 81^{\frac{3}{4}} \) and \( 256^{\frac{3}{4}} \). Starting with \( 81^{\frac{3}{4}} \): \[ 81 = 3^4 \quad \text{thus} \quad 81^{\frac{3}{4}} = (3^4)^{\frac{3}{4}} = 3^{4 \cdot \frac{3}{4}} = 3^3 = 27. \] Now for \( 256^{\frac{3}{4}} \): \[ 256 = 4^4 = (2^2)^4 = 2^8 \quad \text{thus} \quad 256^{\frac{3}{4}} = (2^8)^{\frac{3}{4}} = 2^{8 \cdot \frac{3}{4}} = 2^6 = 64. \] Putting it all together, we have: \[ \frac{81^{\frac{3}{4}}}{256^{\frac{3}{4}}} = \frac{27}{64}. \] Thus, the simplified form of the original expression is: \[ \boxed{\frac{27}{64}}. \]