GED Math: Scientific Notation
Scientific notation shows up in science and unit-conversion problems. To use it, convert between forms as needed: change a number from words (such as “300 million”) to standard form and then to scientific notation, or change an answer choice to standard form to check it. When a conversion factor is a decimal such as 0.907, you can write it in scientific notation as 9.07 × 10−1 and multiply the first factors. Remember that a power of ten with a negative exponent multiplies by a number less than 1. After finding an answer, check that the first factor is between 1 and 10.
1. Powers of Ten and the Idea of Scientific Notation
Scientific notation uses powers of ten to write very large and very small numbers in a short form. A number in scientific notation has two parts: a decimal number that is at least 1 and less than 10, multiplied by a power of ten. For example, 4.4 × 108 is in scientific notation, but 44 × 107 and 0.44 × 109 are not, because 44 and 0.44 are not between 1 and 10. Look for patterns in the powers of ten: with a positive exponent, the exponent equals the number of zeros in standard form (105 = 100,000), and with a negative exponent, the exponent equals the number of decimal places (10−3 = 0.001). Very small numbers use negative powers, and the greater the exponent, the greater the value of 10n.
Worked Example: Which of these is written in scientific notation: 12.5 × 103, 1.25 × 104, or 0.125 × 105? All three equal 12,500.
Only 1.25 × 104, because 1.25 is between 1 and 10. The other two have a first factor of 12.5 and 0.125, which are outside that range.
1. How many zeros are in the standard form of 106?
Explanation: With a positive exponent, the exponent is the number of zeros: 106 = 1,000,000.
2. What is 10−3 in standard form?
Explanation: A negative exponent of 3 means three decimal places: 0.001.
3. Which number is written in scientific notation?
Explanation: The first factor must be at least 1 and less than 10. Only 3.6 qualifies.
4. Which value is the greatest?
Explanation: 104 = 10,000, which is greater than 1, 10, and 0.01.
5. A very small number is expressed in scientific notation. What kind of exponent will the power of ten have?
Explanation: Numbers between 0 and 1 are written with a negative power of ten.
2. Scientific Notation to Standard Form
To change scientific notation to standard form (the ordinary way to write a number), look at the exponent on the 10. If it is positive, move the decimal point that many places to the right, adding zeros as needed. If it is negative, move the decimal point that many places to the left, adding zeros as needed. The number of places you move equals the size of the exponent. A good check: a large exponent should produce a number with many digits, and a negative exponent should produce a small number less than 1.
Worked Example: Write 6.2 × 105 and 3.82 × 10−3 in standard form.
6.2 × 105: move the point 5 places right: 620,000. 3.82 × 10−3: move the point 3 places left: 0.00382.
1. Write 3.7 × 106 in standard form.
Explanation: Move the decimal point 6 places right: 3,700,000.
2. Write 4.8 × 108 in standard form.
Explanation: Move the decimal point 8 places right: 480,000,000.
3. Write 9.16 × 105 in standard form.
Explanation: Move the decimal point 5 places right: 916,000.
4. Write 5.173 × 10−4 in standard form.
Explanation: Move the decimal point 4 places left: 0.0005173.
5. Write 7.2 × 10−3 in standard form.
Explanation: Move the decimal point 3 places left: 0.0072.
6. Write 9.56 × 10−6 in standard form.
Explanation: Move the decimal point 6 places left: 0.00000956.
7. The average distance from Neptune to Earth is 2.67 × 109 miles. What is this in standard form?
Explanation: Move the decimal point 9 places right: 2,670,000,000.
3. Standard Form to Scientific Notation
To write a number in scientific notation, work backward. Step 1: move the decimal point until there is exactly one nonzero digit to its left, so the first factor is between 1 and 10. Step 2: count how many places you moved it. Step 3: multiply by 10 raised to that count. If you moved the point to the left (large number), the exponent is positive; if you moved it to the right (small number), the exponent is negative. Drop trailing zeros from the first factor, since 4.40000000 is just 4.4.
Worked Example: (a) A spacecraft traveled 440,000,000 miles. Write this in scientific notation. (b) A bacterium moves 0.00016 kilometers per hour. Write this in scientific notation.
(a) Move the point 8 places left to get 4.4, so the distance is 4.4 × 108 miles.
(b) Move the point 4 places right to get 1.6, so the speed is 1.6 × 10−4 km/h.
1. Write 2,300 in scientific notation.
Explanation: Move the decimal point 3 places left: 2.3 × 103.
2. Write 12,400,000 in scientific notation.
Explanation: Move the decimal point 7 places left to get 1.24.
3. Write 150,000,000,000 in scientific notation.
Explanation: There are 11 places between the 1 and the end of the number, so the exponent is 11.
4. Write 0.00042 in scientific notation.
Explanation: Move the decimal point 4 places right to get 4.2, so the exponent is −4.
5. Write 0.0095 in scientific notation.
Explanation: Move the point 3 places right to get 9.5, so the exponent is −3.
6. Write 0.00000058 in scientific notation.
Explanation: Move the point 7 places right to get 5.8, so the exponent is −7.
7. Light travels about 300 million meters per second. Write this speed in scientific notation.
Explanation: 300 million = 300,000,000. Moving the point 8 places left gives 3, so 3 × 108.
4. Comparing Numbers in Scientific Notation
You can compare numbers in scientific notation without changing them to standard form. First compare the exponents: for positive numbers, the one with the greater power of ten has the greater value (this is true because each extra power multiplies the value by 10). Only when the exponents are the same do you compare the first factors. This also works for negative exponents: 10−4 is greater than 10−6, because −4 is greater than −6 (0.0001 > 0.000001). On a multiple-choice question whose choices are in scientific notation, change your answer to scientific notation instead of changing every choice.
Worked Example: Which is greater, 4.5 × 103 or 9.8 × 104? Which is greater, 3.2 × 10−4 or 9.1 × 10−5?
Compare exponents: 4 > 3, so 9.8 × 104 is greater (98,000 vs. 4,500). For the second pair, −4 > −5, so 3.2 × 10−4 is greater (0.00032 vs. 0.000091).
1. Which is greater?
Explanation: The power 104 is greater than 103, so 9.8 × 104 is greater.
2. Which is greater?
Explanation: The exponent −4 is greater than −5, so 3.2 × 10−4 is greater (0.00032 > 0.000091).
3. Which is greater?
Explanation: The exponents are the same, so compare the first factors: 4.5 > 3.9.
4. Which list is in order from least to greatest?
Explanation: Order by exponent: −3 < 2 < 5.
5. Which number is the least?
Explanation: The least exponent is −8, so 7.5 × 10−8 is the smallest value.
6. A virus is 1.2 × 10−7 meters wide and a bacterium is 2.5 × 10−6 meters wide. Which is wider, and by about how many times?
Explanation: The exponent −6 is greater than −7, so the bacterium is wider (about 20 times).
5. Applying Scientific Notation to Word Problems
Scientific notation shows up in science and unit-conversion problems. To use it, convert between forms as needed: change a number from words (such as “300 million”) to standard form and then to scientific notation, or change an answer choice to standard form to check it. When a conversion factor is a decimal such as 0.907, you can write it in scientific notation as 9.07 × 10−1 and multiply the first factors. Remember that a power of ten with a negative exponent multiplies by a number less than 1. After finding an answer, check that the first factor is between 1 and 10.
Worked Example: An acre is 43,560 square feet. Write that in scientific notation. A nanosecond is 1.0 × 10−9 second; write it in standard form.
43,560: move the point 4 places left: 4.356 × 104. A nanosecond: move the point 9 places left: 0.000000001 second.
1. An acre is 43,560 square feet. How is this written in scientific notation?
Explanation: Move the decimal point 4 places left to get 4.356.
2. One nanosecond is 1.0 × 10−9 second. Write this in standard form.
Explanation: Move the decimal point 9 places left: 0.000000001.
3. Many satellites orbit about 23,500 miles above Earth. Write the distance in scientific notation.
Explanation: Move the decimal point 4 places left: 2.35 × 104.
4. One ton equals 0.907 metric ton. Which expression gives the weight, in metric tons, of an 8-ton shipment?
Explanation: 0.907 = 9.07 × 10−1 (move the point 1 place right), and 8 tons is 8 times that: 7.256 metric tons.
5. The mass of a dust particle is 7.5 × 10−10 kilograms. Which statement is true?
Explanation: A negative exponent means the number is between 0 and 1, so the mass is far less than 1 kilogram.
6. A population is 3.4 × 106 and a second population is 2.9 × 107. Which statement is true?
Explanation: Divide first factors: 2.9 ÷ 3.4 ≈ 0.85, and the powers differ by 101, so 0.85 × 10 ≈ 8.5. The second population is larger.
Common Mistakes to Avoid
- Writing a first factor outside the range 1 to 10. 12.5 × 103 and 0.125 × 105 are not in scientific notation; the first factor must be at least 1 and less than 10.
- Using the wrong sign on the exponent. Moving the decimal left (large number) gives a positive exponent; moving it right (small number) gives a negative one.
- Moving the decimal point the wrong direction when converting to standard form. A positive exponent moves the point right, a negative exponent moves it left.
- Miscounting the places moved. Count each digit the point passes and compare the number of zeros or decimal places with the exponent.
- Thinking a negative exponent makes the number negative. 10−3 is 0.001, which is positive, just small.
- Comparing the first factors first. Compare the exponents first; only compare first factors when the exponents match.
- Thinking 10−6 is greater than 10−4 because 6 is greater than 4. On the number line, −4 is greater, so 10−4 is the larger value.
- Keeping trailing zeros in the first factor, or dropping zeros that are part of the number. Check by converting back.
Frequently Asked Questions
What is scientific notation?
Scientific notation writes a number as a decimal at least 1 and less than 10, multiplied by a power of ten. For example, 620,000 = 6.2 × 105.
How do I convert scientific notation to standard form?
Move the decimal point the number of places shown by the exponent: to the right if the exponent is positive and to the left if it is negative, adding zeros as needed.
How do I write a small number like 0.00042 in scientific notation?
Move the decimal point to the right until there is one nonzero digit to its left (4.2). You moved 4 places, so the exponent is −4: 4.2 × 10−4.
How do I compare two numbers in scientific notation?
Compare the exponents first; the greater exponent means the greater number (for positive numbers). If the exponents are equal, compare the first factors.
Why does 10 to a negative power give a decimal?
Each negative exponent divides by 10 once more. 10−1 = 0.1, 10−2 = 0.01, and so on, so the exponent counts the decimal places.
When do I use scientific notation on the GED test?
It appears in science and measurement questions with very large or very small numbers, such as astronomical distances or the size of cells, and in questions that ask you to convert or compare them.
Where can I practice more problems like these?
The GED Math practice in the GED Math QBank gives you more practice on this topic, and you can browse every other topic in the full GED Math QBank.