Operations with Scientific Notation
In scientific notation, multiply or divide the first parts and the powers of 10 separately. To add or subtract, match the powers first.
What you'll be able to do
- Multiply and divide numbers in scientific notation.
- Add and subtract them by matching the powers of 10.
- Fix an answer so the first part is from 1 up to 10.
- Solve real problems with huge and tiny numbers.
The big idea
Multiplying. Multiply the first parts. Add the exponents. So (3 × 10⁴)(5 × 10⁶) = 15 × 10¹⁰.
Fixing the first part. 15 is too big. Move the point one place left and add 1 to the exponent. So 15 × 10¹⁰ = 1.5 × 10¹¹. If the first part is below 1, move right and take 1 away. So 0.4 × 10³ = 4 × 10².
Dividing. Divide the first parts. Subtract the exponents. So (8.4 × 10⁷) ÷ (2 × 10³) = 4.2 × 10⁴. Watch the signs: 10⁻³ ÷ 10⁻⁵ = 10², because −3 − (−5) = 2.
Adding and subtracting. The powers of 10 must match first, like adding tens to tens. So 3.2 × 10⁵ + 4 × 10⁴ = 3.2 × 10⁵ + 0.4 × 10⁵ = 3.6 × 10⁵.
Powers and roots. (2 × 10³)² = 2² × 10⁶ = 4 × 10⁶. For a square root, make the exponent even first. So √(9 × 10⁶) = 3 × 10³.
Units. Pick a unit that makes the answer easy to read. 2.5 × 10⁶ bytes is 2.5 megabytes, because a megabyte is 10⁶ bytes.
Words you need
- Scientific notation
- A number from 1 up to 10, times a power of 10. Like 1.5 × 10¹¹.
- Exponent
- The small raised number. In 10¹¹, the exponent is 11.
- Power of 10
- 10 multiplied by itself. 10³ = 1,000 and 10⁻² = 1/100.
- Square root
- The positive number that, times itself, makes a number. √9 = 3.
Which rule when
| To... | First parts | Exponents |
|---|---|---|
| Multiply | Multiply | Add |
| Divide | Divide | Subtract |
| Add or subtract | Add or subtract | Make them match first |
Worked examples
Example 1
Find (3 × 10⁴)(5 × 10⁶). Write the answer in scientific notation.
- Multiply the first parts: 3 × 5 = 15. Add the exponents: 4 + 6 = 10.
- That gives 15 × 10¹⁰.
- 15 is too big, so fix it: 15 × 10¹⁰ = 1.5 × 10¹¹.
Answer: 1.5 × 10¹¹
Example 2
Find (8.4 × 10⁷) ÷ (2 × 10³).
- Divide the first parts: 8.4 ÷ 2 = 4.2.
- Subtract the exponents: 7 − 3 = 4.
- The answer is 4.2 × 10⁴.
Answer: 4.2 × 10⁴
Example 3
Find 3.2 × 10⁵ + 4 × 10⁴.
- Match the powers of 10: 4 × 10⁴ = 0.4 × 10⁵.
- Add the first parts: 3.2 + 0.4 = 3.6.
- The answer is 3.6 × 10⁵.
Answer: 3.6 × 10⁵
Common mistakes
Multiplying the exponents: (2 × 10³)(4 × 10⁵) = 8 × 10¹⁵. Add them. 10³ × 10⁵ = 10⁸, so the answer is 8 × 10⁸.
Adding before the powers match: 5 × 10⁴ + 2 × 10³ = 7 × 10⁴. Match first: 5 × 10⁴ + 0.2 × 10⁴ = 5.2 × 10⁴.
Leaving the first part too big: 42 × 10³. Fix it: 42 × 10³ = 4.2 × 10⁴.
Mixing up signs: 10⁻³ ÷ 10⁻⁵ = 10⁻⁸. Subtract: −3 − (−5) = 2. So the answer is 10².
Practice: 20 questions
12 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Light travels about 3 × 10⁵ km each second. How far does it go in 3 × 10¹ seconds? Word problem
- A company earns 9 × 10⁸ dollars a year from 3 × 10² stores. How much is that per store, on average? Word problem
- One town has 4.2 × 10⁴ people. The next town has 3.5 × 10⁴ people. How many live in both towns? Word problem
- A forest of 8 × 10⁶ trees covers 4 × 10² square kilometers. How many trees grow on each square kilometer? Word problem
- Zoe says (2 × 10³)(4 × 10⁵) = 8 × 10¹⁵. Find her mistake. What is the correct answer?
Tier 2 Intermediate One more step to think through
- A factory fills 6 × 10⁵ bottles a day. Each bottle holds 7 × 10⁻² liters. How many liters is that a day? Word problem
- A tank holds 1.2 × 10⁶ liters. It is shared equally among 4 × 10² homes. How many liters does each home get? Word problem
- One city has 2.3 × 10⁶ people. Another has 9 × 10⁵ people. How many live in both cities? Word problem
- Which is greater, (4 × 10⁵)² or 2 × 10¹¹?
- A phone video is 3.6 × 10⁹ bytes. A gigabyte is 10⁹ bytes. How many gigabytes is the video? Word problem
Tier 3 Advanced Several steps - find the hidden structure
- Light travels 3 × 10⁸ meters per second. The Sun is 1.5 × 10¹¹ meters away. How many minutes does sunlight take to reach Earth, to the nearest tenth? Word problem
- A probe flies 4 × 10⁴ km each hour. It heads to an asteroid 4.8 × 10⁷ km away. How many days does the trip take? Word problem
- Find 0.00042 + 3 × 10⁻⁵. Write the answer in scientific notation.
- A heart pumps about 7 × 10⁻² liters with each beat. It beats about 1 × 10⁵ times a day. How many liters does it pump in 30 days? Word problem
- Lee says (6 × 10⁻³) ÷ (2 × 10⁻⁵) = 3 × 10⁻⁸. Find the mistake. What is the correct answer?
Tier 4 Challenge Stretch problems, some above grade level
- Find (5 × 10⁻⁴)³. Write the answer in scientific notation.
- Find √(1.6 × 10⁹). Write the answer in scientific notation.
- Find the average of 3 × 10⁸ and 5 × 10⁷. Write the answer in scientific notation.
- A heart beats about 1 × 10⁵ times a day. There are about 3.65 × 10² days in a year. About how many times does it beat in 80 years? Round the first part to the nearest tenth. Word problem
- Which is greater, (3 × 10⁴)² or √(1 × 10¹⁸)?
Answers
Tier 1 Foundations
- 9 × 10⁶. 3 × 3 = 9, and 10⁵ × 10¹ = 10⁶ (add the exponents).
- 3 × 10⁶. 9 ÷ 3 = 3, and 10⁸ ÷ 10² = 10⁶ (subtract the exponents).
- 7.7 × 10⁴. 4.2 + 3.5 = 7.7, so the answer is 7.7 × 10⁴.
- 2 × 10⁴. 8 ÷ 4 = 2, and 10⁶ ÷ 10² = 10⁴ (subtract the exponents).
- 8 × 10⁸. Zoe multiplied the exponents. When you multiply, you add them. 2 × 4 = 8, and 10³ × 10⁵ = 10⁸ (add the exponents).
Tier 2 Intermediate
- 4.2 × 10⁴. 6 × 7 = 42, and 10⁵ × 10⁻² = 10³ (add the exponents). Adjust: 42 × 10³ = 4.2 × 10⁴.
- 3 × 10³. 1.2 ÷ 4 = 0.3, and 10⁶ ÷ 10² = 10⁴ (subtract the exponents). Adjust: 0.3 × 10⁴ = 3 × 10³.
- 3.2 × 10⁶. Match the powers: 9 × 10⁵ = 0.9 × 10⁶. 2.3 + 0.9 = 3.2, so the answer is 3.2 × 10⁶.
- 2 × 10¹¹ is greater. In scientific notation, 2 × 10¹¹ is greater than (4 × 10⁵)² = 1.6 × 10¹¹.
- 3.6. Divide: (3.6 × 10⁹) ÷ (1 × 10⁹) = 3.6.
Tier 3 Advanced
- about 8.3 minutes. (1.5 × 10¹¹) ÷ (3 × 10⁸) = 0.5 × 10³ = 500 seconds. 500 ÷ 60 is about 8.3 minutes.
- 50 days. (4.8 × 10⁷) ÷ (4 × 10⁴) = 1.2 × 10³ = 1,200 hours. 1,200 ÷ 24 = 50 days.
- 4.5 × 10⁻⁴. 0.00042 = 4.2 × 10⁻⁴. And 3 × 10⁻⁵ = 0.3 × 10⁻⁴. Add the first parts: 4.2 + 0.3 = 4.5. So the sum is 4.5 × 10⁻⁴.
- 2.1 × 10⁵. In one day: 7 × 1 = 7, and 10⁻² × 10⁵ = 10³. That is 7 × 10³ liters. In 30 days: 7 × 10³ × 30 = 210 × 10³ = 2.1 × 10⁵ liters.
- 3 × 10². Lee added the exponents. Dividing subtracts them: −3 − (−5) = 2. 6 ÷ 2 = 3, and 10⁻³ ÷ 10⁻⁵ = 10² (subtract the exponents).
Tier 4 Challenge
- 1.25 × 10⁻¹⁰. Cube each part. 5³ = 125, and (10⁻⁴)³ = 10⁻¹². So the answer is 125 × 10⁻¹² = 1.25 × 10⁻¹⁰.
- 4 × 10⁴. 10⁹ has an odd exponent, so rewrite first: 1.6 × 10⁹ = 16 × 10⁸. Then √16 = 4 and √(10⁸) = 10⁴. So the answer is 4 × 10⁴.
- 1.75 × 10⁸. Match the powers: 5 × 10⁷ = 0.5 × 10⁸. The sum is 3.5 × 10⁸. Half of that is 1.75 × 10⁸.
- about 2.9 × 10⁹ beats. In one year: 1 × 10⁵ × 3.65 × 10² = 3.65 × 10⁷ beats. In 80 years: 3.65 × 10⁷ × 80 = 292 × 10⁷ = 2.92 × 10⁹. That rounds to 2.9 × 10⁹.
- √(1 × 10¹⁸) is greater. (3 × 10⁴)² = 9 × 10⁸. And √(10¹⁸) = 10⁹, which is 10 × 10⁸. So the square root is greater.
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