1. **Stating the problem:** We have four numbers in standard form: $1.5 \times 10^4$, $4.8 \times 10^6$, $3.2 \times 10^2$, and $7.5 \times 10^3$. We need to find: (a) The smallest product possible when multiplying any two of these numbers. (b) The largest sum possible when adding any two of these numbers. All answers must be given in standard form. 2. **Formula and rules:** - Multiplying numbers in standard form: $$ (a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n} $$ - Adding numbers in standard form requires the powers of 10 to be the same. We convert one number to match the other's power of 10 before adding. 3. **Step (a): Smallest product** Calculate all products: - $(1.5 \times 10^4) \times (4.8 \times 10^6) = (1.5 \times 4.8) \times 10^{4+6} = 7.2 \times 10^{10}$ - $(1.5 \times 10^4) \times (3.2 \times 10^2) = (1.5 \times 3.2) \times 10^{4+2} = 4.8 \times 10^{6}$ - $(1.5 \times 10^4) \times (7.5 \times 10^3) = (1.5 \times 7.5) \times 10^{4+3} = 11.25 \times 10^{7} = 1.125 \times 10^{8}$ (converted to standard form) - $(4.8 \times 10^6) \times (3.2 \times 10^2) = (4.8 \times 3.2) \times 10^{6+2} = 15.36 \times 10^{8} = 1.536 \times 10^{9}$ - $(4.8 \times 10^6) \times (7.5 \times 10^3) = (4.8 \times 7.5) \times 10^{6+3} = 36 \times 10^{9} = 3.6 \times 10^{10}$ - $(3.2 \times 10^2) \times (7.5 \times 10^3) = (3.2 \times 7.5) \times 10^{2+3} = 24 \times 10^{5} = 2.4 \times 10^{6}$ The smallest product is $2.4 \times 10^{6}$ from $(3.2 \times 10^2) \times (7.5 \times 10^3)$. 4. **Step (b): Largest sum** Check sums of pairs: - $(7.5 \times 10^3) + (1.5 \times 10^4)$ Convert $1.5 \times 10^4$ to $15 \times 10^3$ to add: $$7.5 \times 10^3 + 15 \times 10^3 = (7.5 + 15) \times 10^3 = 22.5 \times 10^3 = 2.25 \times 10^4$$ - $(7.5 \times 10^3) + (4.8 \times 10^6)$ Convert $7.5 \times 10^3$ to $0.0075 \times 10^6$: $$0.0075 \times 10^6 + 4.8 \times 10^6 = (0.0075 + 4.8) \times 10^6 = 4.8075 \times 10^6$$ - $(7.5 \times 10^3) + (3.2 \times 10^2)$ Convert $3.2 \times 10^2$ to $0.32 \times 10^3$: $$7.5 \times 10^3 + 0.32 \times 10^3 = (7.5 + 0.32) \times 10^3 = 7.82 \times 10^3$$ - $(4.8 \times 10^6) + (3.2 \times 10^2)$ Convert $3.2 \times 10^2$ to $0.00032 \times 10^6$: $$4.8 \times 10^6 + 0.00032 \times 10^6 = (4.8 + 0.00032) \times 10^6 = 4.80032 \times 10^6$$ - $(1.5 \times 10^4) + (3.2 \times 10^2)$ Convert $3.2 \times 10^2$ to $0.032 \times 10^4$: $$1.5 \times 10^4 + 0.032 \times 10^4 = (1.5 + 0.032) \times 10^4 = 1.532 \times 10^4$$ - $(1.5 \times 10^4) + (4.8 \times 10^6)$ Convert $1.5 \times 10^4$ to $0.015 \times 10^6$: $$0.015 \times 10^6 + 4.8 \times 10^6 = (0.015 + 4.8) \times 10^6 = 4.815 \times 10^6$$ The largest sum is $4.815 \times 10^6$ from $(1.5 \times 10^4) + (4.8 \times 10^6)$. **Final answers:** - Smallest product: $\boxed{2.4 \times 10^{6}}$ - Largest sum: $\boxed{4.815 \times 10^{6}}$