1. **Simplify using product law** (i) Simplify $\left(\frac{5}{4}\right)^2 \times \left(\frac{3}{7}\right)^4$ - The product law states: $a^m \times a^n = a^{m+n}$ only if bases are the same. - Here, bases differ, so multiply directly: $$\left(\frac{5}{4}\right)^2 = \frac{25}{16}, \quad \left(\frac{3}{7}\right)^4 = \frac{81}{2401}$$ - Multiply fractions: $$\frac{25}{16} \times \frac{81}{2401} = \frac{2025}{38416}$$ (ii) Simplify $p^6 \times p^7$ - Same base $p$, so add exponents: $$p^{6+7} = p^{13}$$ (iii) Simplify $\left(\frac{3}{4}\right)^6 \times \left(\frac{1}{5}\right)^6$ - Same exponent, multiply bases: $$\left(\frac{3}{4} \times \frac{1}{5}\right)^6 = \left(\frac{3}{20}\right)^6$$ (iv) Simplify $s^5 \times x^4 \times y^3 \times s^5$ - Combine like bases: $$s^{5+5} \times x^4 \times y^3 = s^{10} x^4 y^3$$ 2. **Identify base and exponent, find value** (i) $5^4$ - Base: 5, Exponent: 4 - Value: $5 \times 5 \times 5 \times 5 = 625$ (ii) $2^5$ - Base: 2, Exponent: 5 - Value: $2 \times 2 \times 2 \times 2 \times 2 = 32$ (iii) $6^3$ - Base: 6, Exponent: 3 - Value: $6 \times 6 \times 6 = 216$ (iv) $\left(\frac{2}{3}\right)^6$ - Base: $\frac{2}{3}$, Exponent: 6 - Value: $\left(\frac{2}{3}\right)^6 = \frac{2^6}{3^6} = \frac{64}{729}$ 3. **Convert into rational numbers** (i) 0.2 = $\frac{1}{5}$ (ii) 4.8 = $\frac{24}{5}$ (iii) 6.0 = $\frac{6}{1}$ (iv) 0.75 = $\frac{3}{4}$