8. \( \sqrt[4]{8 a^{3} b} \cdot \sqrt[4]{10 a^{2} b^{7}} \)

To simplify the expression \( \sqrt[4]{8 a^{3} b} \cdot \sqrt[4]{10 a^{2} b^{7}} \), we can first combine the two roots: \[ \sqrt[4]{8 a^{3} b} \cdot \sqrt[4]{10 a^{2} b^{7}} = \sqrt[4]{(8 a^{3} b) \cdot (10 a^{2} b^{7})} \] Now, multiply the terms inside the root: \[ 8 \cdot 10 = 80, \quad a^{3} \cdot a^{2} = a^{5}, \quad b \cdot b^{7} = b^{8} \] Thus, we have: \[ \sqrt[4]{80 a^{5} b^{8}} \] Now we can simplify the fourth root: \[ \sqrt[4]{80} \cdot \sqrt[4]{a^{5}} \cdot \sqrt[4]{b^{8}} \] Now simplify each part: 1. **For \( \sqrt[4]{80} \)**: 80 can be factored as \( 16 \cdot 5 \), so \( \sqrt[4]{80} = \sqrt[4]{16 \cdot 5} = \sqrt[4]{16} \cdot \sqrt[4]{5} = 2 \cdot \sqrt[4]{5} \). 2. **For \( \sqrt[4]{a^{5}} \)**: This can be expressed as \( a^{5/4} = a^{1} \cdot a^{1/4} \). 3. **For \( \sqrt[4]{b^{8}} \)**: This simplifies easily since \( \sqrt[4]{b^{8}} = b^{8/4} = b^{2} \). Putting it all together, we have: \[ \sqrt[4]{80 a^{5} b^{8}} = 2 \sqrt[4]{5} \cdot a^{1} a^{1/4} \cdot b^{2} \] So the final simplified expression is: \[ 2b^{2} a \sqrt[4]{5} a^{1/4} \] Thus, we can express this result more clearly as: \[ 2 a^{1 + \frac{1}{4}} b^{2} \sqrt[4]{5} = 2 a^{\frac{5}{4}} b^{2} \sqrt[4]{5} \] So this is the simplified expression: \[ 2 a^{\frac{5}{4}} b^{2} \sqrt[4]{5} \]