Subjects algebra

Simplify Force Expression 70069B

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1. **Stating the problem:** Simplify the expression $$Fy = -10 - 9 \times 10^{-9} \cdot r \cdot \frac{3 \cdot r_0 y}{4 \pi \cdot 10^{-9} \times 10^{-4}} \div (36 \pi)$$. 2. **Rewrite the expression clearly:** $$Fy = -10 - 9 \times 10^{-9} \cdot r \cdot \frac{3 r_0 y}{4 \pi \times 10^{-9} \times 10^{-4}} \times \frac{1}{36 \pi}$$ 3. **Simplify the denominator inside the fraction:** $$4 \pi \times 10^{-9} \times 10^{-4} = 4 \pi \times 10^{-13}$$ 4. **Rewrite the fraction:** $$\frac{3 r_0 y}{4 \pi \times 10^{-13}} = 3 r_0 y \times \frac{1}{4 \pi \times 10^{-13}} = \frac{3 r_0 y}{4 \pi} \times 10^{13}$$ 5. **Substitute back:** $$Fy = -10 - 9 \times 10^{-9} \cdot r \cdot \frac{3 r_0 y}{4 \pi} \times 10^{13} \times \frac{1}{36 \pi}$$ 6. **Combine constants:** $$9 \times 10^{-9} \times 10^{13} = 9 \times 10^{4}$$ 7. **Rewrite:** $$Fy = -10 - r \cdot \frac{3 r_0 y}{4 \pi} \cdot \frac{9 \times 10^{4}}{36 \pi}$$ 8. **Simplify the fraction with constants:** $$\frac{9 \times 10^{4}}{36 \pi} = \frac{9}{36} \times \frac{10^{4}}{\pi} = \frac{1}{4} \times \frac{10^{4}}{\pi} = \frac{10^{4}}{4 \pi}$$ 9. **Substitute:** $$Fy = -10 - r \cdot \frac{3 r_0 y}{4 \pi} \times \frac{10^{4}}{4 \pi} = -10 - r \cdot 3 r_0 y \times \frac{10^{4}}{16 \pi^{2}}$$ 10. **Final simplified form:** $$Fy = -10 - \frac{3 \times 10^{4}}{16 \pi^{2}} \cdot r r_0 y$$