1. **Stating the problem:** We have four numbers: $1.5 \times 10^4$, $4.8 \times 10^6$, $3.2 \times 10^2$, and $7.5 \times 10^3$. (a) Find the smallest product of any two numbers. (b) Find the largest sum of any two numbers. 2. **Formula and rules:** - To multiply numbers in scientific notation: multiply the coefficients and add the exponents. - To add numbers in scientific notation, convert to the same power of 10 or to standard form. 3. **Part (a) smallest product:** Calculate products of pairs: - $(1.5 \times 10^4) \times (4.8 \times 10^6) = (1.5 \times 4.8) \times 10^{4+6} = 7.2 \times 10^{10}$ - $(1.5 \times 10^4) \times (3.2 \times 10^2) = (1.5 \times 3.2) \times 10^{4+2} = 4.8 \times 10^{6}$ - $(1.5 \times 10^4) \times (7.5 \times 10^3) = (1.5 \times 7.5) \times 10^{4+3} = 11.25 \times 10^{7} = 1.125 \times 10^{8}$ (rewriting coefficient) - $(4.8 \times 10^6) \times (3.2 \times 10^2) = (4.8 \times 3.2) \times 10^{6+2} = 15.36 \times 10^{8} = 1.536 \times 10^{9}$ - $(4.8 \times 10^6) \times (7.5 \times 10^3) = (4.8 \times 7.5) \times 10^{6+3} = 36 \times 10^{9} = 3.6 \times 10^{10}$ - $(3.2 \times 10^2) \times (7.5 \times 10^3) = (3.2 \times 7.5) \times 10^{2+3} = 24 \times 10^{5} = 2.4 \times 10^{6}$ The smallest product is $2.4 \times 10^{6}$ from $(3.2 \times 10^2) \times (7.5 \times 10^3)$. 4. **Part (b) largest sum:** Convert to standard form: - $1.5 \times 10^4 = 15000$ - $4.8 \times 10^6 = 4800000$ - $3.2 \times 10^2 = 320$ - $7.5 \times 10^3 = 7500$ Calculate sums: - $15000 + 4800000 = 4815000$ - $15000 + 320 = 15320$ - $15000 + 7500 = 22500$ - $4800000 + 320 = 4800320$ - $4800000 + 7500 = 4807500$ - $7500 + 320 = 7820$ The largest sum is $4815000$ from $1.5 \times 10^4 + 4.8 \times 10^6$. **Final answers:** - (a) Smallest product: $2.4 \times 10^{6}$ - (b) Largest sum: $4.815 \times 10^{6}$ (rewriting $4815000$ in scientific notation)