Scientific Notation Relations Cf6771

1. **State the problem:** We need to find the relationship between each pair of numbers given in scientific notation. 2. **Recall the rule:** To compare numbers in scientific notation $a \times 10^m$ and $b \times 10^n$, divide one by the other: $$\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}$$ 3. **Compare each pair:** - Pair 1: $7 \times 10^{-4}$ and $7 \times 10^{-2}$ $$\frac{7 \times 10^{-2}}{7 \times 10^{-4}} = \frac{7}{7} \times 10^{-2 - (-4)} = 1 \times 10^{2} = 100$$ So, second is 100 times larger than first. - Pair 2: $2 \times 10^{-3}$ and $5 \times 10^{-3}$ $$\frac{5 \times 10^{-3}}{2 \times 10^{-3}} = \frac{5}{2} \times 10^{-3 - (-3)} = 2.5 \times 10^{0} = 2.5$$ So, second is 2.5 times larger than first, or equivalently, first is 2.5 times smaller than second. - Pair 3: $4 \times 10^{8}$ and $4 \times 10^{9}$ $$\frac{4 \times 10^{9}}{4 \times 10^{8}} = \frac{4}{4} \times 10^{9 - 8} = 1 \times 10^{1} = 10$$ So, second is 10 times larger than first. - Pair 4: $3 \times 10^{7}$ and $6 \times 10^{5}$ $$\frac{3 \times 10^{7}}{6 \times 10^{5}} = \frac{3}{6} \times 10^{7 - 5} = 0.5 \times 10^{2} = 0.5 \times 100 = 50$$ So, first is 50 times larger than second. 4. **Match each pair with the correct relationship:** - $7 \times 10^{-4}$ and $7 \times 10^{-2}$: Second is 100 times larger than first. - $2 \times 10^{-3}$ and $5 \times 10^{-3}$: First is 2.5 times smaller than second. - $4 \times 10^{8}$ and $4 \times 10^{9}$: Second is 10 times larger than first. - $3 \times 10^{7}$ and $6 \times 10^{5}$: First is 50 times larger than second. **Final answer:** - Pair 1: Second is 100 times larger than first. - Pair 2: First is 2.5 times smaller than second. - Pair 3: Second is 10 times larger than first. - Pair 4: First is 50 times larger than second.