HighSchoolQ

➕ Algebra 1 · Exponents and Polynomials

Scientific Notation

Write huge and tiny numbers as a front number times a power of 10, convert both ways, compare, multiply, divide, add and subtract, with diagrams and practice.

Grades 9, 10⏱ 15 min✔ AI-checkedCCSS 8.EE.A.3CCSS 8.EE.A.4
About standards and checking

Written by AI, then independently re-solved and checked by separate AI reviewers and automated answer checks. Written by Claude (Sonnet 5.5) from the concept template. Automated gate: schema, notation and every math answer verified by evaluated expressions. Two blind re-solves by separate AI solvers with no access to the answer key: every keyed answer matched. Two prose audits by a separate, stronger AI reviewer (Claude Opus 5.5): every finding fixed, no wrong answer keys found. Final small edits re-checked by the automated gate (no human teacher review yet). Last checked 2026-11-08.

  • 8.EE.A.3: Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.
  • 8.EE.A.4: Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used.

By the end you can…

  • Write large and small positive numbers in scientific notation, convert back to ordinary form, and explain why the exponent is positive or negative.
  • Compare two numbers in scientific notation and say how many times as large one is as the other.
  • Multiply and divide numbers in scientific notation, fixing the front number when it leaves the range from 1 up to (but not including) 10.
  • Add and subtract numbers in scientific notation by matching the powers of 10 first.
  • Use scientific notation for real sizes and read calculator output such as 3.2E5.

Lesson (school style)

Scientific notation is a short way to write very large or very small numbers: a front number that is at least 1 and less than 10, times a power of 10. The speed of light is about 300,000,000 meters per second, which is easier to write as about 3.0 × 108 meters per second. The exponent keeps track of how far the decimal point has slid. The rules for multiplying and dividing come straight from the laws of exponents you already know. All numbers in this lesson are positive.

The form: front number times a power of 10

A positive number is in scientific notation when it is written as a × 10n, where

  • the front number a is at least 1 and less than 10 (written 1 ≤ a < 10), so it has exactly one nonzero digit to the left of the decimal point, and
  • the exponent n is an integer: positive, negative or zero.

For example, 4.5 × 106 is in scientific notation. The numbers 45 × 105 and 0.45 × 107 have the same value, but they are not in scientific notation, because 45 is too big and 0.45 is too small to be the front number. The number 10 itself is 1 × 101. Zero cannot be written this way, because a × 10n is always more than 0.

A zero at the end of a front number, as in 3.0, does not change its value. Scientists sometimes keep it to show how precisely a measurement is known.

Why powers of 10? Our place-value system is built on 10: each place is worth 10 times as much as the place to its right. So a power of 10 records how far a decimal point has slid. Remember 103 = 1,000, and from the last lesson, 10−3 = 1/103 = 1/1,000 = 0.001.

Big numbers: slide left, exponent positive

To write 2,600,000 in scientific notation:

  1. Find the decimal point. In a whole number it sits at the far right: 2,600,000.
  2. Slide it left until exactly one nonzero digit is to its left: 2.6.
  3. Count the slides: 6.
  4. Write 2.6 × 106.

Why is the exponent positive? Sliding the point 6 places left made the front number 1,000,000 times smaller (2,600,000 became 2.6). To keep the value the same, you must multiply by 1,000,000 again, and that is 106. So 2.6 × 106 = 2.6 × 1,000,000 = 2,600,000. Each slide left makes the front number 10 times smaller, so each slide adds 1 to the exponent.

Count the slides, not the zeros. The number 2,600,000 has five zeros, but the exponent is 6.

Small numbers: slide right, exponent negative

To write 0.0000071 in scientific notation, slide the point right until exactly one nonzero digit is to its left: 7.1. The point passes five zeros and the 7, so that is 6 slides.

Why is the exponent negative? Sliding right made the front number 1,000,000 times bigger (0.0000071 became 7.1). To keep the value the same, you must divide by 1,000,000. That is multiplying by 10−6 = 1/1,000,000. So 0.0000071 = 7.1 × 10−6. Each slide right makes the front number 10 times bigger, so each slide subtracts 1 from the exponent.

A quick sign check for positive numbers:

  • A number that is 10 or more has a positive exponent.
  • A number from 1 up to (but not including) 10 has exponent 0, because 100 = 1. For example, 7.1 × 100 = 7.1.
  • A number between 0 and 1 has a negative exponent.

The number itself stays positive. In scientific notation, a negative exponent means the number is between 0 and 1, not that the number is negative.

Back to ordinary form

Read the exponent as a slide instruction. A positive exponent slides the point right that many places, and a negative exponent slides it left. Fill any empty places with zeros.

  • 9.2 × 104: slide 4 places right, giving 92,000.
  • 9.2 × 10−4: slide 4 places left, giving 0.00092.

If the front number has more digits than the exponent needs, the point lands inside the digits: 4.567 × 102 = 456.7. As a size check, a positive exponent should give a number of 10 or more, and a negative exponent should give a number between 0 and 1.

Comparing numbers and “how many times as much”

To compare two positive numbers in scientific notation, compare the exponents first. The bigger exponent wins. If the exponents are equal, compare the front numbers. For example, 8 × 105 = 800,000 is smaller than 2.1 × 106 = 2,100,000, even though 8 is bigger than 2.1. With negative exponents, remember that −2 is greater than −4, so 3 × 10−2 is greater than 9 × 10−4 (0.03 is greater than 0.0009).

Why does the exponent win? A front number is less than 10, so a number with exponent n is less than 10n+1. A number with exponent n + 1 is at least 10n+1. So a number with exponent n + 1 is always bigger than a number with exponent n, and the same idea works for bigger gaps between exponents.

This only works when both numbers are in scientific notation. Look at 45 × 105 and 2 × 106. The exponents seem to say 2 × 106 is bigger, but 45 × 105 = 4.5 × 106 is bigger. Fix the front numbers first.

To say how many times as large one number is as another, divide the larger by the smaller. For 9 × 108 and 3 × 105, we get (9 × 108) ÷ (3 × 105) = 3 × 103, so the first is 3,000 times as large.

Multiplying and dividing

Multiply the front numbers and add the exponents:

(a × 10m)(b × 10n) = (a × b) × 10m+n.

Why? Multiplication can be done in any order, so group the front numbers together and the powers of 10 together. The product rule from Laws of Exponents says 10m × 10n = 10m+n, and it still works when an exponent is negative.

Divide the front numbers and subtract the exponents:

(a × 10m) ÷ (b × 10n) = (a ÷ b) × 10m−n.

This is the quotient rule. (Here b is at least 1, so it is never 0.) Be careful when the second exponent is negative, because subtracting a negative adds: 105 ÷ 10−3 = 105−(−3) = 108.

Fix the front number. The product or quotient of the front numbers can land outside the range from 1 up to (but not including) 10. Then slide its decimal point and adjust the exponent the way you did before: each slide left adds 1, and each slide right subtracts 1. For example, 18 × 107 = 1.8 × 108, and 0.25 × 106 = 2.5 × 105.

Adding and subtracting: match the powers first

Adding is different from multiplying. If both numbers have the same power of 10, add the front numbers and keep the power:

a × 10n + b × 10n = (a + b) × 10n.

This is the distributive property read backward, the same move as combining like terms: 3x + 5x = 8x. Here 10n plays the role of x.

If the powers are different, rewrite one number so they match. Slide the point of its front number and adjust the exponent so the value stays the same. For example, 7 × 102 = 0.7 × 103. Either power works. Then add or subtract the front numbers, keep the power, and fix the front number if it left the range from 1 up to (but not including) 10.

Do not add the exponents when you add. A power of 10 acts like a unit, and hundred-thousands and ten-thousands are different-sized units. You must match the units before you add the front numbers. For instance, 105 + 104 = 100,000 + 10,000 = 110,000, not 109.

Real sizes, units and calculators

Scientific notation is made for measurements at extreme sizes. These facts are all approximate, so we say “about”:

  • Light travels about 3.0 × 108 meters per second.
  • The Sun is about 1.5 × 108 kilometers from Earth.
  • A human hair is about 1 × 10−4 meters wide.

Keep the units straight. You can only add or subtract quantities that use the same unit. Before you multiply or divide, check that the units fit together, and convert if they do not. A kilometer is 103 meters, and a millimeter is 10−3 meters. Choosing a unit of a sensible size keeps numbers short: a hair that is 1 × 10−4 meters wide is about 0.1 millimeters wide.

Calculators and spreadsheets often show scientific notation as E notation. A display of 7.9E5 means 7.9 × 105 = 790,000, and 5.9E-3 means 5.9 × 10−3 = 0.0059. The E means “times ten to the power.” It is not the special number 2.718... that you may meet in later math courses.

Key words

Scientific notation
A way to write a number as a × 10n, where 1 ≤ a < 10 and n is an integer. This lesson uses positive numbers; a negative number just gets a minus sign in front, as in −4.5 × 106.
Front number
The number a in a × 10n (also called the coefficient). It is at least 1 and less than 10, with one nonzero digit to the left of the decimal point.
Power of 10
An expression such as 103 = 1,000 or 10−2 = 1/100. In scientific notation, its exponent counts how many places the decimal point slid.
Exponent
The small raised number in a power. In scientific notation, for a positive number it is positive when the number is 10 or more, 0 when the number is from 1 up to (but not including) 10, and negative when the number is between 0 and 1.
Ordinary form
A number written the usual way with digits and a decimal point, like 92,000 or 0.00092. It is also called decimal notation. Some books call it standard form, but on this site standard form means something else (for example, a polynomial in descending order).
E notation
Calculator shorthand for scientific notation. The display 3.2E5 means 3.2 × 105.

Worked examples

Example 1. Write 7,310,000 in scientific notation.

  1. The decimal point of a whole number is at the far right: 7,310,000.
  2. Slide the point left until one nonzero digit is to its left: 7.31. The point passes 0, 0, 0, 0, 1 and 3, so that is 6 slides.
  3. The number is 10 or more, so the exponent is positive: 6.
  4. Write it: 7.31 × 106.
  5. Check: 7.31 × 1,000,000 = 7,310,000.

Answer: Answer: 7.31 × 106

Example 2. Write 0.000094 in scientific notation.

  1. Slide the point right until one nonzero digit is to its left: 9.4. The point passes 0, 0, 0, 0 and 9, so that is 5 slides.
  2. Sliding right made the front number bigger, so the power of 10 has to divide to pay it back. The number is between 0 and 1, so the exponent is negative: −5.
  3. Write it: 9.4 × 10−5.
  4. Check: 10−5 = 1/100,000 = 0.00001, and 9.4 × 0.00001 = 0.000094.

Answer: Answer: 9.4 × 10−5

Example 3. Write 6.25 × 105 in ordinary form.

  1. The exponent is positive (5), so slide the decimal point 5 places to the right.
  2. The first two slides pass the digits 2 and 5. The other three slides need zeros: 625,000.
  3. Size check: 6.25 is about 6, and 105 = 100,000, so the answer should be about 600,000.

Answer: Answer: 625,000

Example 4. Write 7.4 × 10−3 in ordinary form.

  1. The exponent is negative (−3), so slide the decimal point 3 places to the left, filling in zeros: 7.4 → 0.74 → 0.074 → 0.0074.
  2. Check: 10−3 = 1/1,000, and 7.4 ÷ 1,000 = 0.0074.
  3. Size check: the answer is positive and between 0 and 1, as a negative exponent should give.

Answer: Answer: 0.0074

Example 5. Multiply (5.5 × 106)(4 × 10−2). Give the answer in scientific notation.

  1. Multiply the front numbers: 5.5 × 4 = 22.
  2. Add the exponents: 6 + (−2) = 4.
  3. So far the product is 22 × 104, but 22 is not between 1 and 10.
  4. Slide the point of 22 one place left (22 → 2.2) and add 1 to the exponent: 2.2 × 105.
  5. Check with ordinary numbers: 5,500,000 × 0.04 = 220,000 = 2.2 × 105.

Answer: Answer: 2.2 × 105

Example 6. Divide (2.4 × 10−3) ÷ (6 × 10−7). Give the answer in scientific notation.

  1. Divide the front numbers: 2.4 ÷ 6 = 0.4.
  2. Subtract the exponents: −3 − (−7) = −3 + 7 = 4.
  3. So far the quotient is 0.4 × 104, but 0.4 is less than 1.
  4. Slide the point one place right (0.4 → 4) and subtract 1 from the exponent: 4 × 103.
  5. Check: 0.0024 ÷ 0.0000006 = 4,000.

Answer: Answer: 4 × 103

Example 7. Add 3.2 × 105 + 6.8 × 104. Give the answer in scientific notation.

  1. The powers of 10 are different (5 and 4), so we cannot add the front numbers yet.
  2. Rewrite the second number with 105: sliding the point of 6.8 one place left gives 0.68, so add 1 to the exponent. 6.8 × 104 = 0.68 × 105.
  3. Now the powers match: 3.2 × 105 + 0.68 × 105 = (3.2 + 0.68) × 105.
  4. Add the front numbers and keep the power: 3.88 × 105. The front number 3.88 is between 1 and 10, so nothing needs fixing.
  5. Check: 320,000 + 68,000 = 388,000 = 3.88 × 105.

Answer: Answer: 3.88 × 105

Example 8. Subtract 8.1 × 10−3 − 6.4 × 10−4. Give the answer in scientific notation.

  1. The exponents differ (−3 and −4). Rewrite the second number with 10−3: sliding the point of 6.4 one place left gives 0.64, so add 1 to the exponent. 6.4 × 10−4 = 0.64 × 10−3.
  2. Subtract the front numbers: 8.1 − 0.64 = 7.46.
  3. Keep the power: 7.46 × 10−3.
  4. Check: 0.0081 − 0.00064 = 0.00746 = 7.46 × 10−3.

Answer: Answer: 7.46 × 10−3

Example 9. Sunlight travels about 3.0 × 108 meters per second. The Sun is about 1.5 × 108 kilometers from Earth, and 1 kilometer is 103 meters. About how many seconds does sunlight take to reach Earth?

  1. Fix the units first. The speed uses meters but the distance uses kilometers, so convert the distance to meters. Since a kilometer is 103 meters, 1.5 × 108 km = 1.5 × 108 × 103 m = 1.5 × 1011 m.
  2. Time = distance ÷ speed, so the time is (1.5 × 1011) ÷ (3.0 × 108) seconds.
  3. Divide the front numbers: 1.5 ÷ 3.0 = 0.5. Subtract the exponents: 11 − 8 = 3. So the quotient is 0.5 × 103.
  4. Fix the front number: 0.5 → 5 and subtract 1 from the exponent, giving 5 × 102 seconds.
  5. That is about 500 seconds, or a little over 8 minutes (500 ÷ 60 is about 8.3).

Answer: Answer: About 5 × 102 seconds, or a little over 8 minutes

Example 10. Which is larger, 6 × 10−3 or 2 × 10−5? How many times as large is it?

  1. Compare the exponents first: −3 is greater than −5, so 6 × 10−3 is larger. (In ordinary form, 0.006 is greater than 0.00002.)
  2. To find how many times as large, divide the larger by the smaller: (6 × 10−3) ÷ (2 × 10−5).
  3. Divide the front numbers: 6 ÷ 2 = 3. Subtract the exponents: −3 − (−5) = 2. So the quotient is 3 × 102 = 300.
  4. Check: 0.006 ÷ 0.00002 = 300.

Answer: Answer: 6 × 10−3 is larger. It is 300 times as large.

In plain English

Some numbers are huge, like the distance to the Sun. Some are tiny, like the width of a hair. Writing all those zeros is slow, and it is easy to lose count. Scientific notation is a shortcut.

You write the number as a front number times a power of 10. The front number has to be at least 1 and less than 10. The power of 10 tells you how far the decimal point slid. To write 8,900,000, slide the point 6 places left to get 8.9. So it is 8.9 × 106.

Big number? The point slides left and the exponent is positive. Small number, between 0 and 1? The point slides right and the exponent is negative. A negative exponent does not make the number negative. It only means the number is between 0 and 1.

To multiply, multiply the front numbers and add the exponents. To divide, divide the front numbers and subtract the exponents. If the front number ends up 10 or more, or less than 1, slide its point and fix the exponent.

To add or subtract, first make the two powers of 10 match. Then add or subtract the front numbers and keep the power. Do not add the exponents when you add.

See it

Count the slides: the direction gives the sign of the exponent Big number: 7,310,000 6 slides left 7 3 1 0 0 0 0 1 2 3 4 5 6 = 7.31 × 106 6 slides left, so exponent +6 (a number of 10 or more) Small number: 0.000094 5 slides right 0 0 0 0 0 9 4 1 2 3 4 5 = 9.4 × 10−5 5 slides right, so exponent −5 (a number between 0 and 1) Hollow dot: where the decimal point starts. Solid dot: where it stops. Each slide left makes the front number 10 times smaller, so the exponent goes up by 1. Each slide right makes it 10 times bigger, so the exponent goes down by 1.
Count the slides of the decimal point. Sliding left (a big number) gives a positive exponent, and sliding right (a number below 1) gives a negative exponent: 7,310,000 = 7.31 × 106 and 0.000094 = 9.4 × 10−5.
Adding: match the powers of 10, then add the front numbers 3.2 × 105 + 6.8 × 104 point 1 place left, exponent up by 1 1. Match the powers 3.2 × 105 + 0.68 × 105 same power of 10 2. Add the front numbers (3.2 + 0.68) × 105 = 3.88 × 105 Check: 320,000 + 68,000 = 388,000 = 3.88 × 105 Do not add the exponents: 105 + 104 is 110,000, not 109.
Step 1: match the powers of 10. Step 2: add the front numbers and keep the power. 3.2 × 105 + 6.8 × 104 = 3.88 × 105.

The same idea, the fun way

Pick the one that clicks. Each one explains the same concept from a different angle.

Gaming

Idle-game scoreboards

In some idle and clicker games, the score grows so large that the game stops showing every digit and switches to something like 3.4e8. That is the game’s way of printing scientific notation.

How it connects: The display 3.4e8 means 3.4 × 108, which is 340,000,000. The e stands for “times ten to the power,” and it is not the special number 2.718... from later math courses. Limit: a game or calculator display may round the front number, so 3.4e8 can be only an approximation of the exact score.

Music

The range of hearing

People with typical hearing can hear sounds from about 20 vibrations per second (hertz) up to about 20,000 hertz. The exact range varies from person to person and shrinks with age. Musical notes sit inside this range: the lowest key on a standard 88-key piano vibrates about 27.5 times per second.

How it connects: In scientific notation, that range is about 2 × 101 to 2 × 104 hertz. Dividing gives (2 × 104) ÷ (2 × 101) = 1 × 103, so the highest sounds people hear vibrate about 1,000 times as fast as the lowest. Limit: the exponents differ by 3, and that gave exactly 1,000 only because both front numbers are 2. When the front numbers differ, you must divide them too.

Movie

Alice in Wonderland (the book and its films)

In Lewis Carroll’s Alice’s Adventures in Wonderland, and the films based on it, Alice drinks from a bottle labeled DRINK ME and shrinks, and later eats something labeled EAT ME and grows. Pretend that each sip of the shrinking drink makes her one tenth as tall as before. After 3 sips she is 10−3 times her starting height.

How it connects: Shrinking gives a negative exponent, and growing would give a positive one. A negative exponent here means “a fraction of the original size,” not a negative size. An Alice who starts at 1.6 meters tall would end up 1.6 × 10−3 meters tall, which is 1.6 millimeters. Limit: the one-tenth-per-sip rule is made up for this picture. The story has no such rule.

Idiom

Get on the same page

To “get on the same page” means to make sure everyone shares the same understanding before moving on.

How it connects: Before you add or subtract numbers in scientific notation, get them on the same page: the same power of 10. Then 4 × 102 + 7 × 101 becomes 4 × 102 + 0.7 × 102 = 4.7 × 102, which matches 400 + 70 = 470. Limit: this is only for adding and subtracting. When you multiply or divide, the powers do not have to match.

Sports

Marathon versus sprint

A 100-meter dash is 1 × 102 meters long. A marathon is 42,195 meters, which is about 4.2 × 104 meters.

How it connects: To see how many times as long the marathon is, divide: (4.2 × 104) ÷ (1 × 102) = 4.2 × 102, which is about 420 times as long. Limit: the exponents differ by 2, but the answer is not just 100 times, because the front numbers (4.2 and 1) matter too. The exponent difference only gives the power of 10 in the answer.

Nature

The ladder of sizes

A typical atom is about 1 × 10−10 meters across, and a human hair is about 1 × 10−4 meters wide.

How it connects: How many times as wide is the hair? Divide: (1 × 10−4) ÷ (1 × 10−10) = 10−4−(−10) = 106, so the hair is about a million times as wide as a typical atom. Notice that −4 − (−10) is +6, not −14. Limit: both sizes are rounded to one digit, and atoms vary in size, so “about a million” is a rough comparison, not an exact count.

Food

Reading a nutrition label

Suppose a snack label says it has 450 milligrams of sodium. A milligram is 10−3 grams.

How it connects: So 450 milligrams is 450 × 10−3 grams, which is 0.45 grams, or 4.5 × 10−1 grams in scientific notation. Changing units changes the numbers we write but not the amount. Limit: 450 × 10−3 is not in scientific notation, because 450 is not less than 10. Also, convert to one unit before you add or subtract amounts, and check that the units fit together before you multiply or divide.

Everyday joke

The step counter

Why did the decimal point wear a step counter? In scientific notation, the exponent counts every step it takes.

How it connects: Slide the decimal point of 0.00083 right 4 places to get 8.3, and you get 8.3 × 10−4. The size of the exponent counts the steps, and the sign records the direction: right for a number below 1 gives a negative exponent, and left for a number of 10 or more gives a positive one. Limit: it counts the steps of the point, not the zeros.

Memory help

Acronym

MOVE

To convert a number to scientific notation: Move the decimal point until exactly one nonzero digit is to its left. Observe how many places it moved: that count is the size of the exponent. Verify the direction: sliding left (a number of 10 or more) gives a positive exponent, and sliding right (a number between 0 and 1) gives a negative exponent. Express the answer as the front number times the power of 10.

Rhyme

Where the exponent sits

Ten or more? The exponent is above zero. Under one? It dips below. From one up to ten, it sits at zero, and the slides you count show how far you go. (This is for positive numbers.)

Mnemonic

Match, Add, Fix

For adding or subtracting numbers in scientific notation: Match the powers of 10 (rewrite one number), Add or subtract the front numbers and keep the power, then Fix the front number if it left the range from 1 up to (but not including) 10. Multiplying and dividing skip the Match step, because their powers do not have to match. They use the level-down rule from Laws of Exponents: multiplying adds exponents and dividing subtracts them.

Picture it

Draw the hops

Write the number and draw one small arc for every place the decimal point slides, from where it starts to where it stops. Number the arcs 1, 2, 3 and so on. The last number is the size of the exponent, and the direction of the arcs gives the sign: left is positive, right is negative. Counting arcs keeps you from counting zeros by mistake.

Story

The tidy-up crew

After you multiply or divide, picture a tidy-up crew checking the front number. If it is 10 or more, the crew slides its point left and tells the exponent to go up. If it is less than 1, the crew slides the point right and tells the exponent to go down. Each slide changes the front number by a factor of 10, and the exponent changes to make up for it, so the value never changes.

Trap alert

Count the slides, not the zeros

The number 1,700,000 has five zeros, but it is 1.7 × 106, not 1.7 × 105. The exponent is the number of places the decimal point slides, not the number of zeros. For a number with a decimal like 0.009, count the slides: past 0, 0 and 9 is 3 slides.

Study notes

Key ideas

  • Scientific notation is a × 10n with 1 ≤ a < 10 and n an integer. All numbers here are positive.
  • The exponent counts how many places the decimal point slides. Sliding left (a number of 10 or more) gives a positive exponent. Sliding right (a number between 0 and 1) gives a negative exponent.
  • For positive numbers in scientific notation, compare the exponents first. If they are equal, compare the front numbers. To find how many times as large, divide larger by smaller.
  • Multiply: multiply the fronts and add the exponents. Divide: divide the fronts and subtract the exponents. Then fix the front number if it left the range from 1 up to (but not including) 10.
  • Add or subtract: match the powers of 10 first, then add or subtract the fronts and keep the power.

Steps

  1. To convert: slide the decimal point until one nonzero digit is on its left, count the slides, and set the sign by the direction.
  2. To multiply or divide: work on the front numbers and on the exponents separately.
  3. To add or subtract: rewrite one number so both use the same power of 10.
  4. Fix the front number so it is at least 1 and less than 10.
  5. Check with a size estimate or with ordinary numbers, and keep the units straight.

Formulas

  • Scientific notation: a × 10n with 1 ≤ a < 10 and n an integer
  • Multiply: (a × 10m)(b × 10n) = (a × b) × 10m+n
  • Divide: (a × 10m) ÷ (b × 10n) = (a ÷ b) × 10m−n
  • Add: a × 10n + b × 10n = (a + b) × 10n (same exponent)
  • Subtract: a × 10n − b × 10n = (a − b) × 10n (same exponent)
  • How many times as large: (larger number) ÷ (smaller number)

Watch out

  • Count the slides of the decimal point, not the zeros: 1,700,000 has five zeros, but it equals 1.7 × 106.
  • A negative exponent does not make a negative number: 2 × 10−3 = 0.002, a small positive number.
  • When multiplying, add the exponents (do not multiply them). When adding, do not add the exponents: match the powers, then add the front numbers.
  • After multiplying or dividing, check the front number. 22 × 104 and 0.4 × 104 are not finished.
  • Subtracting a negative exponent adds: 5 − (−3) = 8, not 2.
  • Check the units: convert so they fit together before you add, subtract, multiply or divide.

Why it works

  • Why does the exponent count the slides? Each slide of the decimal point changes the front number by a factor of 10, and the power of 10 undoes it, so the value stays the same.
  • Why add exponents when multiplying? Reorder the factors so the two powers of 10 sit together. 10m × 10n uses m factors of 10 and then n more factors of 10, which is m + n factors of 10 in all, and the product rule keeps working when an exponent is negative.
  • Why match the powers before adding? It is the same as combining like terms. 3x + 5x = 8x works because both terms count the same unit (x). In the same way, 3 × 105 + 5 × 105 = 8 × 105 counts the same unit (105).

Common mistakes

✗ MistakeThe exponent is the number of zeros: 1,700,000 = 1.7 × 105.
✓ InsteadCount the places the decimal point slides, not the zeros. 1,700,000 → 1.7 slides 6 places, so 1,700,000 = 1.7 × 106.
✗ MistakeA negative exponent makes the number negative: 8 × 10−4 = −8,000.
✓ InsteadA negative exponent means dividing by a power of 10: 8 × 10−4 = 8 ÷ 10,000 = 0.0008, a small positive number.
✗ MistakeTo add, add the exponents: 3 × 102 + 4 × 102 = 7 × 104.
✓ InsteadWhen the powers match, add the front numbers and keep the power: 3 × 102 + 4 × 102 = 7 × 102. Only multiplying adds exponents.
✗ MistakeTo multiply, multiply the exponents: (2 × 103)(4 × 105) = 8 × 1015.
✓ InsteadMultiply the front numbers and add the exponents: (2 × 103)(4 × 105) = 8 × 108.
✗ Mistake38 × 106 is fine, because the value is right.
✓ InsteadThe front number must be at least 1 and less than 10. Slide the point left and raise the exponent: 38 × 106 = 3.8 × 107.

Teach it back

Explain to a friend how to write 0.0000725 in scientific notation, why its exponent is negative, and how to multiply it by 4 × 103.

Explaining it out loud shows what you really know. Did you…

  • I slid the point 5 places right to get 7.25 and wrote 7.25 × 10−5.
  • I said the exponent is negative because sliding right made the front number 105 times bigger, so the power of 10 has to divide it back. The number itself is still positive.
  • I multiplied the fronts (7.25 × 4 = 29) and added the exponents (−5 + 3 = −2), then fixed the front number: 29 × 10−2 = 2.9 × 10−1.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

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Practice it your way

Flashcards

Flip, recall and space your reviews.

Cheat sheet

One page. Everything you need.

True or false

Fast yes/no questions with reasons.

Matching game

Pair them up against the clock.

Practice questions

Step-up questions with explanations.

Laws of ExponentsZero and Negative ExponentsCombining Like Terms