Scientific notation writes a number as a value between 1 and 10, multiplied by a power of 10 — useful for very large or very small numbers that would otherwise need a long string of zeros. The number 8,200,000 in scientific notation is 8.2 × 10⁶. The number 0.000047 is 4.7 × 10⁻⁵.
To convert a large number into scientific notation, move the decimal point left until only one nonzero digit remains before it, and count how many places you moved — that count becomes the positive exponent. For 8,200,000: moving the decimal point from the end (8200000.) left six places gives 8.2, so 8,200,000 = 8.2 × 10⁶. To convert a small decimal, move the decimal point right until one nonzero digit sits before it, and the count becomes a negative exponent. For 0.000047: moving the decimal point right five places gives 4.7, so 0.000047 = 4.7 × 10⁻⁵.
Converting back to standard notation reverses the process: a positive exponent means move the decimal point right that many places (padding with zeros as needed); a negative exponent means move it left. 3.05 × 10⁴ becomes 30,500 (moving the decimal four places right). 6.1 × 10⁻³ becomes 0.0061 (moving three places left).
Worked example one: write 152,000,000 in scientific notation. Moving the decimal from the end left until one digit remains: 1.52 × 10⁸. Worked example two: convert 9.4 × 10⁻² back to standard form. Moving the decimal two places left from 9.4: 0.094.
A common mistake is leaving more than one digit before the decimal point in the "coefficient" — writing 82 × 10⁵ instead of the correct standard form 8.2 × 10⁶ (the two are mathematically equal, since 82 × 10⁵ = 8.2 × 10 × 10⁵ = 8.2 × 10⁶, but only the second is proper scientific notation). Another common mistake is getting the exponent's sign backwards — using a positive exponent for a small number or a negative exponent for a large one, when the rule is: large numbers (greater than or equal to 10) get positive exponents, and small numbers (between 0 and 1) get negative exponents.
Scientific notation is the standard way extremely large and small real-world measurements are actually written in science: the speed of light is roughly 3 × 10⁸ meters per second, and the mass of an electron is roughly 9.1 × 10⁻³¹ kilograms — both numbers that would be unwieldy and error-prone to write out in full standard notation with dozens of zeros. Engineering notation, a close cousin, restricts the exponent to multiples of 3 (matching the metric system's kilo-, mega-, milli-, micro- prefixes) rather than requiring exactly one digit before the decimal point, which is why you'll sometimes see 82 × 10³ instead of 8.2 × 10⁴ in an engineering context specifically — both are valid within their respective conventions, but they're not interchangeable without knowing which convention a document is using.
Scientific notation is especially useful for keeping track of significant figures precisely and for multiplying or dividing very large or very small numbers without losing track of place value — multiplying (3 × 10⁴) × (2 × 10³) is just (3 × 2) × 10^(4+3) = 6 × 10⁷, using the same "add exponents when multiplying same-base powers" rule covered on the Exponent Calculator. Adding or subtracting numbers in scientific notation requires an extra step multiplying and dividing don't: the exponents must match before you can combine the coefficients. Adding (3 × 10⁴) + (5 × 10³) isn't simply 8 × 10⁴ or 8 × 10³ — you first rewrite one term to share the other's exponent (5 × 10³ = 0.5 × 10⁴), then add the coefficients: 3 × 10⁴ + 0.5 × 10⁴ = 3.5 × 10⁴. Skipping the exponent-matching step is the single most common error when combining scientific-notation values by hand.
For more worked examples and the reasoning behind the rule, see What Is Scientific Notation?