Exponents & roots
Scientific Notation
Scientific notation writes a number as a value between 1 and 10 multiplied by a power of ten. So 93,000,000 becomes 9.3 × 10⁷ and 0.000052 becomes 5.2 × 10⁻⁵. The exponent counts how many places the decimal point moved. This guide shows how to convert in both directions, how to multiply, divide, add and subtract in scientific notation, and how to read the “E” format on a calculator.
What is scientific notation?
Scientific notation is a compact way to write very large and very small numbers. Every number takes the same shape:
The coefficient must have exactly one non-zero digit before the decimal point. 9.3 × 10⁷ is in scientific notation; 93 × 10⁶ and 0.93 × 10⁸ have the same value but are not, because 93 is too big and 0.93 too small.
The exponent does the work that a long string of zeros would otherwise do. A positive exponent means a big number (multiply by 10 that many times); a negative exponent means a small number (divide by 10 that many times); an exponent of zero means the number is between 1 and 10, since 10⁰ = 1. The exponent rules explain why 10⁻⁵ means 1 ÷ 10⁵.
Two reasons it is used everywhere in science and engineering:
- Size at a glance. Comparing 6.02 × 10²³ with 9.11 × 10⁻³¹ takes one look at the exponents. Counting the zeros in 602,000,000,000,000,000,000,000 does not.
- Precision is explicit. Every digit in the coefficient is significant, so
1.20 × 10³clearly has three significant figures where 1200 is ambiguous. The sig fig rules rely on this.
How to convert a number to scientific notation
- Move the decimal point until exactly one non-zero digit is to its left.
- Count the places you moved it. That count is the size of the exponent.
- Choose the sign. If the original number was 10 or larger, the exponent is positive. If it was smaller than 1, the exponent is negative.
- Drop leading and trailing zeros that only held places, unless they are significant.
Example: a large number, 93,000,000
93 million miles is roughly the average distance from Earth to the Sun, which is where you will often see this number.
Example: a small number, 0.000052
Example: a number that is already close, 45,600 and 0.0089
A memory aid: the exponent tells you how many times you would multiply 1 by 10 to reach the same scale. A large number needs many multiplications, so a large positive exponent. A tiny number needs divisions, so a negative one. The place value chart makes the place counting visual.
How to convert scientific notation to standard form
Reverse the process. The exponent tells you how far and which way to move the decimal point.
- Positive exponent: move the point to the right that many places, adding zeros as needed.
- Negative exponent: move the point to the left that many places, adding zeros in front.
3.4 × 10⁻⁷ is a tiny positive number. A negative number has a minus sign on the coefficient: −3.4 × 10⁷ is −34,000,000.Multiplying and dividing in scientific notation
These are the easy operations, because powers of ten combine by adding or subtracting exponents.
Example: (3 × 10⁴) × (5 × 10³)
The tidy step is where most errors happen. Moving the decimal point one place left in the coefficient adds one to the exponent, so the value is unchanged.
Example: (7.5 × 10⁻³) × (4 × 10⁻⁵)
Example: (6.4 × 10⁵) ÷ (1.6 × 10²)
Example: (2.4 × 10⁻³) ÷ (6 × 10⁵)
When the coefficient drops below 1, moving the point right subtracts one from the exponent.
Example: how long sunlight takes to reach Earth
The average Earth–Sun distance is about 1.496 × 10¹¹ meters, and light travels about 2.998 × 10⁸ meters per second.
To check the coefficient arithmetic on long numbers, the multiplication calculator and division calculator show every step, and the big number calculator handles the full standard-form values exactly.
Adding and subtracting in scientific notation
You can only add or subtract coefficients when the powers of ten match, the same way you can only add like terms in algebra. So the first step is to rewrite one number to match the other’s exponent.
- Rewrite the number with the smaller exponent so its exponent matches the larger one.
- Add or subtract the coefficients.
- Keep the shared power of ten, and tidy the coefficient if needed.
Example: 3.2 × 10⁵ + 4.5 × 10⁴
Check in standard form: 320,000 + 45,000 = 365,000. It matches.
Example: 8.1 × 10⁶ − 2.4 × 10⁵
3 × 10⁴ + 2 × 10⁴ is 5 × 10⁴, not 5 × 10⁸. Exponents add only when you multiply.Powers and roots of numbers in scientific notation
To raise a number in scientific notation to a power, raise the coefficient to that power and multiply the exponent by it.
For a square root, halve the exponent. That only works cleanly when the exponent is even, so adjust it first if it is odd.
The square root calculator gives √90 to as many places as you need.
E notation on calculators and spreadsheets
Calculators, spreadsheets and programming languages often show scientific notation with the letter E, which stands for “times ten to the.”
| E notation | Scientific notation | Standard form |
|---|---|---|
| 6.02E23 | 6.02 × 10²³ | 602,000,000,000,000,000,000,000 |
| 1.5E8 | 1.5 × 10⁸ | 150,000,000 |
| 4.56E4 | 4.56 × 10⁴ | 45,600 |
| 3.4E-7 | 3.4 × 10⁻⁷ | 0.00000034 |
| 2E-3 | 2 × 10⁻³ | 0.002 |
When you type a number into a calculator in E form, use the dedicated EE or EXP key rather than typing “× 10 ^”. Typing 1 ÷ 6.02 × 10 ^ 23 follows the order of operations: it divides 1 by 6.02 and then multiplies by 10²³, which is not what you meant. The EE key, or parentheses, keeps the number together as a single value. The order of operations calculator shows how an expression like this is evaluated.
Engineering notation
Engineering notation is a variant where the exponent is always a multiple of 3, so the coefficient ranges from 1 to 999. It lines up with metric prefixes and with the words thousand, million and billion: 45,600 = 45.6 × 10³ (45.6 thousand) and 0.000052 = 52 × 10⁻⁶ (52 millionths).
Powers of ten reference table
| Power of ten | Standard form | Name (US) | Metric prefix |
|---|---|---|---|
| 10¹² | 1,000,000,000,000 | one trillion | tera (T) |
| 10⁹ | 1,000,000,000 | one billion | giga (G) |
| 10⁶ | 1,000,000 | one million | mega (M) |
| 10³ | 1,000 | one thousand | kilo (k) |
| 10² | 100 | one hundred | hecto (h) |
| 10¹ | 10 | ten | deca (da) |
| 10⁰ | 1 | one | — |
| 10⁻¹ | 0.1 | one tenth | deci (d) |
| 10⁻² | 0.01 | one hundredth | centi (c) |
| 10⁻³ | 0.001 | one thousandth | milli (m) |
| 10⁻⁶ | 0.000001 | one millionth | micro (µ) |
| 10⁻⁹ | 0.000000001 | one billionth | nano (n) |
Some familiar quantities in scientific notation: the speed of light is about 3.00 × 10⁸ meters per second, Avogadro’s number is about 6.022 × 10²³, and the mass of an electron is about 9.109 × 10⁻³¹ kilograms.
Common mistakes
- Leaving the coefficient outside 1 to 10. 15 × 10⁷ and 0.4 × 10⁻⁸ are correct values but not finished answers. Tidy them to 1.5 × 10⁸ and 4 × 10⁻⁹.
- Moving the exponent the wrong way when tidying. Making the coefficient smaller makes the exponent bigger, and vice versa. The value must not change.
- Reading a negative exponent as a negative number. 10⁻³ is 0.001, a positive number.
- Adding exponents when adding numbers. Match the exponents first, then add only the coefficients.
- Miscounting places. 0.000052 needs 5 moves, not 4 or 6. Count the jumps to just past the first non-zero digit, or check by converting back. Errors with negatives here are easy; the negative number rules help when subtracting exponents such as
−3 − 5 = −8. - Dropping significant zeros. If a measurement is 1.20 × 10³, the zero is meaningful; writing 1.2 × 10³ claims less precision.
Frequently asked questions
How do you write 5,000,000 in scientific notation?
5,000,000 is 5 × 10⁶. Move the decimal point from the end of the number six places to the left so that only the 5 is in front of it. The number was large, so the exponent is positive 6. In E notation on a calculator it appears as 5E6.
How do you write 0.00034 in scientific notation?
0.00034 is 3.4 × 10⁻⁴. Move the decimal point four places to the right, past the first non-zero digit 3, giving 3.4. Because the original number was smaller than 1, the exponent is negative: −4. To check, move the point four places left from 3.4 and you get back 0.00034.
What are the rules for scientific notation?
Write the number as a × 10ⁿ, where the coefficient a is at least 1 and less than 10, and n is a whole number. Numbers of 10 or more get a positive exponent, numbers between 0 and 1 get a negative exponent, and numbers from 1 up to 10 use 10⁰. Every digit in the coefficient is significant.
How do you multiply numbers in scientific notation?
Multiply the coefficients and add the exponents. For (3 × 10⁴)(5 × 10³), the coefficients give 15 and the exponents give 10⁷, so the result is 15 × 10⁷. Then tidy the coefficient into the 1-to-10 range: 15 becomes 1.5 and the exponent goes up by one, giving 1.5 × 10⁸.
What does E mean on a calculator?
E means “times ten to the power of.” A display of 6.02E23 means 6.02 × 10²³, and 3.4E-7 means 3.4 × 10⁻⁷, or 0.00000034. Calculators switch to this format when a result has too many digits to fit on the screen. Enter such numbers with the EE or EXP key.
Can the exponent in scientific notation be zero?
Yes. An exponent of zero means the number is already between 1 and 10, because 10⁰ = 1. For example, 7.25 in scientific notation is 7.25 × 10⁰. In practice the × 10⁰ is often left off, but it is correct to include it when every value in a table should have the same format.
How do you add numbers in scientific notation?
Rewrite them so the powers of ten match, then add the coefficients and keep the shared power. For 3.2 × 10⁵ + 4.5 × 10⁴, rewrite the second as 0.45 × 10⁵. Adding gives 3.65 × 10⁵, which equals 365,000. Never add the exponents when adding numbers.
Calculators for this
Every one of these shows the working, so you can check the method as well as the answer.
More guides
https://fourfunctioncalculator.com/scientific-notation/