To calculate with numbers in scientific notation, use the exponent rules on the powers of 10 and ordinary arithmetic on the first factors.
Why it works
Multiplying: since multiplication can be regrouped, $(3 \times 10^{4})(2 \times 10^{5}) = (3 \cdot 2) \times (10^{4} \cdot 10^{5}) = 6 \times 10^{9}$.
Adding is different. $3 \times 10^{4} + 2 \times 10^{5}$ is 3 ten-thousands plus 2 hundred-thousands. You can only add amounts of the same unit, just as you line up place values, so first rewrite one of them: $2 \times 10^{5} = 20 \times 10^{4}$.
Adding is different. $3 \times 10^{4} + 2 \times 10^{5}$ is 3 ten-thousands plus 2 hundred-thousands. You can only add amounts of the same unit, just as you line up place values, so first rewrite one of them: $2 \times 10^{5} = 20 \times 10^{4}$.
Key idea
Multiply or divide: work on the first factors, and add or subtract the exponents. Add or subtract: first make the powers of 10 match, then add or subtract the first factors and keep the power. Finally, adjust so the first factor is between 1 and 10.
Worked examples
Worked example: Multiply, then adjust
Find $(5 \times 10^{3})(4 \times 10^{2})$.
- First factors: $5 \cdot 4 = 20$. Powers: $10^{3} \cdot 10^{2} = 10^{5}$.
- $20 \times 10^{5}$ is not in scientific notation, since $20 \ge 10$.
- $20 = 2 \times 10^{1}$, so the product is $2 \times 10^{1} \times 10^{5} = 2 \times 10^{6}$.
Answer: $2 \times 10^{6}$
Worked example: Divide
Find $\dfrac{8.4 \times 10^{7}}{2 \times 10^{3}}$.
- First factors: $8.4 \div 2 = 4.2$.
- Powers: $10^{7-3} = 10^{4}$.
Answer: $4.2 \times 10^{4}$
Worked example: Add
Find $3.2 \times 10^{5} + 4 \times 10^{4}$.
- Make the powers match: $4 \times 10^{4} = 0.4 \times 10^{5}$.
- Add the first factors and keep $10^{5}$: $(3.2 + 0.4) \times 10^{5}$.
Answer: $3.6 \times 10^{5}$
Worked example: Divide, then adjust
Find $\dfrac{1.5 \times 10^{-2}}{3 \times 10^{4}}$.
- First factors: $1.5 \div 3 = 0.5$. Powers: $10^{-2-4} = 10^{-6}$.
- $0.5$ is less than 1, so write $0.5 = 5 \times 10^{-1}$.
- $5 \times 10^{-1} \times 10^{-6} = 5 \times 10^{-7}$.
Answer: $5 \times 10^{-7}$
Common mistake
Adding the exponents when adding: $(2 \times 10^{3}) + (3 \times 10^{3}) = 5 \times 10^{6}$. Check in standard form: $2{,}000 + 3{,}000 = 5{,}000 = 5 \times 10^{3}$. Adding exponents is the rule for multiplying.