Simplify \( \frac{16^{\frac{3}{4}} \times 4^{\frac{1}{2}}}{8^{\frac{5}{3}}} \)

To simplify \( \frac{16^{\frac{3}{4}} \times 4^{\frac{1}{2}}}{8^{\frac{5}{3}}} \), we start by expressing all the numbers in terms of base 2: 1. **Breaking down the bases**: - \( 16 = 2^4 \), so \( 16^{\frac{3}{4}} = (2^4)^{\frac{3}{4}} = 2^{4 \cdot \frac{3}{4}} = 2^3 = 8 \). - \( 4 = 2^2 \), so \( 4^{\frac{1}{2}} = (2^2)^{\frac{1}{2}} = 2^{2 \cdot \frac{1}{2}} = 2^1 = 2 \). - \( 8 = 2^3 \), so \( 8^{\frac{5}{3}} = (2^3)^{\frac{5}{3}} = 2^{3 \cdot \frac{5}{3}} = 2^5 = 32 \). 2. **Substituting back into the expression**: Now we have: \[ \frac{16^{\frac{3}{4}} \times 4^{\frac{1}{2}}}{8^{\frac{5}{3}}} = \frac{2^3 \times 2}{2^5} \] 3. **Combining the numerator**: The numerator \( 2^3 \times 2 = 2^{3+1} = 2^4 \). 4. **Final simplification**: Now we simplify: \[ \frac{2^4}{2^5} = 2^{4-5} = 2^{-1} = \frac{1}{2} \] So, the simplified expression is \( \frac{1}{2} \).