h) \( \left(\frac{1}{16}\right)^{\frac{3}{4}}+4^{\frac{3}{4}} \cdot 8^{\frac{1}{2}} \)

Let’s break down the expression \( \left(\frac{1}{16}\right)^{\frac{3}{4}} + 4^{\frac{3}{4}} \cdot 8^{\frac{1}{2}} \). First, we'll simplify \( \left(\frac{1}{16}\right)^{\frac{3}{4}} \): \[ \frac{1}{16} = 16^{-1} = (2^4)^{-1} = 2^{-4} \] So: \[ \left(\frac{1}{16}\right)^{\frac{3}{4}} = (2^{-4})^{\frac{3}{4}} = 2^{-4 \cdot \frac{3}{4}} = 2^{-3} = \frac{1}{8} \] Next, simplify \( 4^{\frac{3}{4}} \cdot 8^{\frac{1}{2}} \): \[ 4^{\frac{3}{4}} = (2^2)^{\frac{3}{4}} = 2^{2 \cdot \frac{3}{4}} = 2^{\frac{3}{2}} = 2^{1.5} = 2\sqrt{2} \] And: \[ 8^{\frac{1}{2}} = (2^3)^{\frac{1}{2}} = 2^{3 \cdot \frac{1}{2}} = 2^{\frac{3}{2}} = 2\sqrt{2} \] Combining these: \[ 4^{\frac{3}{4}} \cdot 8^{\frac{1}{2}} = (2\sqrt{2})(2\sqrt{2}) = 4 \cdot 2 = 8 \] Now we can combine both parts of the expression: \[ \frac{1}{8} + 8 \] To add, we can convert \( 8 \) to a fraction: \[ 8 = \frac{64}{8} \] Thus: \[ \frac{1}{8} + \frac{64}{8} = \frac{1 + 64}{8} = \frac{65}{8} \] Therefore, the final result is: \[ \boxed{\frac{65}{8}} \]