1. **State the problem:** We are given two equations: $$0 = r - x s^{10} + \frac{8}{x^2} s^{15}$$ $$0 = 8 - x n + x^2 s^{15}$$ and an approximation: $$(4 + 2x - 3 - 2x s^{15}) \approx x s^{15}$$ 2. **Analyze the first equation:** $$0 = r - x s^{10} + \frac{8}{x^2} s^{15}$$ We want to isolate $r$: $$r = x s^{10} - \frac{8}{x^2} s^{15}$$ 3. **Analyze the second equation:** $$0 = 8 - x n + x^2 s^{15}$$ Isolate $n$: $$x n = 8 + x^2 s^{15}$$ Divide both sides by $x$: $$n = \frac{8 + x^2 s^{15}}{x}$$ Show cancellation: $$n = \frac{\cancel{8} + x^2 s^{15}}{\cancel{x}}$$ (Note: 8 and $x$ do not cancel, so no cancellation here; just division) 4. **Approximation given:** $$(4 + 2x - 3 - 2x s^{15}) \approx x s^{15}$$ Simplify left side: $$4 - 3 + 2x - 2x s^{15} = 1 + 2x - 2x s^{15}$$ So: $$1 + 2x - 2x s^{15} \approx x s^{15}$$ Bring all terms to one side: $$1 + 2x - 2x s^{15} - x s^{15} \approx 0$$ Combine like terms: $$1 + 2x - 3x s^{15} \approx 0$$ 5. **Summary:** - From the first equation, $r = x s^{10} - \frac{8}{x^2} s^{15}$ - From the second equation, $n = \frac{8 + x^2 s^{15}}{x}$ - Approximation simplifies to $1 + 2x - 3x s^{15} \approx 0$ These expressions relate the variables $r$, $n$, $x$, and $s$ with powers of $s$. **Final answers:** $$r = x s^{10} - \frac{8}{x^2} s^{15}$$ $$n = \frac{8 + x^2 s^{15}}{x}$$ $$1 + 2x - 3x s^{15} \approx 0$$