What an exponent is
An exponent indicates how many times a base should be multiplied by itself. In 2³ (2 to the power of 3), the base is 2 and the exponent is 3, meaning 2 × 2 × 2, which equals 8. Exponents are a compact way to represent repeated multiplication, and show up in formulas for population growth, compound interest, computer science, and physics.
Negative exponents
A negative exponent represents the reciprocal of the power with the corresponding positive exponent: x⁻ⁿ = 1 / xⁿ. For example, 2 to the power of -3 equals 1 divided by 2 to the power of 3, which is 1/8, or 0.125. This rule holds for any non-zero base.
Fractional exponents
An exponent in fraction form represents a root: x^(1/n) = ⁿ√x. For example, 8 to the power of 1/3 is the same as the cube root of 8, which is 2. An exponent like 3/2 combines both ideas: raise to the numerator first, then take the root matching the denominator, or the other way around, the result is the same. Note that a negative base raised to a fractional exponent doesn't always have a real result, which is why this calculator flags it when that happens.
What scientific notation is
Scientific notation is a way to write very large or very small numbers compactly, representing the number as a coefficient between 1 and 10 multiplied by a power of 10. For example, the approximate distance from Earth to the Sun, 149,600,000 kilometers, can be written as 1.496 × 10⁸. A very small number, like 0.0000456, is written as 4.56 × 10⁻⁵.
Worked example
Converting 1,234,000 to scientific notation: move the decimal point until only one digit remains before it (1.234), counting how many places it moved, 6 places to the left in this case, which gives the exponent. The result is 1.234 × 10⁸. To convert back, simply move the decimal point 6 places to the right, recovering the original number.