Simplify Expression 69187A

1. **State the problem:** Simplify the expression $$\left(\frac{5de^{7}}{2e^{3}}\right) \times \left(\frac{4d^{2}ef^{17}}{def^{6}}\right)$$. 2. **Write the expression clearly:** $$\frac{5de^{7}}{2e^{3}} \times \frac{4d^{2}ef^{17}}{def^{6}}$$ 3. **Simplify each fraction separately:** - In the first fraction, simplify the $e$ terms: $$\frac{5de^{7}}{2e^{3}} = \frac{5d \cancel{e^{7}}^{e^{3} \times e^{4}}}{2 \cancel{e^{3}}} = \frac{5d e^{4}}{2}$$ - In the second fraction, simplify the $d$, $e$, and $f$ terms: $$\frac{4d^{2}ef^{17}}{def^{6}} = \frac{4 d^{2} e f^{17}}{d e f^{6}} = 4 \frac{\cancel{d^{2}}^{d \times d}}{\cancel{d}} \frac{\cancel{e}}{\cancel{e}} \frac{f^{17}}{f^{6}} = 4 d f^{11}$$ 4. **Multiply the simplified fractions:** $$\left(\frac{5d e^{4}}{2}\right) \times (4 d f^{11}) = \frac{5 d e^{4}}{2} \times 4 d f^{11}$$ 5. **Multiply numerators and denominators:** $$= \frac{5 d e^{4} \times 4 d f^{11}}{2} = \frac{20 d^{2} e^{4} f^{11}}{2}$$ 6. **Simplify the fraction by dividing numerator and denominator by 2:** $$= \frac{\cancel{20}^{10} d^{2} e^{4} f^{11}}{\cancel{2}^{1}} = 10 d^{2} e^{4} f^{11}$$ **Final answer:** $$10 d^{2} e^{4} f^{11}$$