1. **Problem (a): Simplify \(\frac{x^2 + 7x}{x^2 - 49}\)** 2. First, factor both numerator and denominator: \[x^2 + 7x = x(x + 7)\] \[x^2 - 49 = (x - 7)(x + 7)\] 3. Substitute the factored forms: \[\frac{x(x + 7)}{(x - 7)(x + 7)}\] 4. Cancel the common factor \(x + 7\): \[\frac{x\cancel{(x + 7)}}{(x - 7)\cancel{(x + 7)}} = \frac{x}{x - 7}\] 5. **Final simplified form:** \(\frac{x}{x - 7}\) --- 6. **Problem (b)(i): Find \(r\) such that \(x^2 \times x^6 = x^r\)** 7. Use the rule for multiplying powers with the same base: add exponents \[x^2 \times x^6 = x^{2 + 6} = x^8\] 8. So, \(r = 8\) --- 9. **Problem (b)(ii): Find \(s\) such that \(s^3 = 8\)** 10. Take the cube root of both sides: \[s = \sqrt[3]{8} = 2\] --- **Summary:** - (a) Simplified expression: \(\frac{x}{x - 7}\) - (b)(i) \(r = 8\) - (b)(ii) \(s = 2\)