Exponent & Power Calculator
Raise a number to any power, find the nth root, or convert to and from scientific notation — with step-by-step working.
Laws of exponents
An exponent tells you how many times to multiply a number (the base) by itself. The power bn means b multiplied n times. These laws let you simplify expressions quickly:
aᵐ × aⁿ = aᵐ⁺ⁿ. Add the exponents when multiplying same bases. e.g. 2³ × 2² = 2⁵ = 32.
aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Subtract the exponents when dividing same bases. e.g. 5⁵ ÷ 5² = 5³ = 125.
(aᵐ)ⁿ = aᵐⁿ. Multiply the exponents. e.g. (3²)³ = 3⁶ = 729.
a⁰ = 1 for any a ≠ 0. e.g. 7⁰ = 1, 1000⁰ = 1.
a⁻ⁿ = 1 / aⁿ. The reciprocal. e.g. 2⁻³ = 1/8 = 0.125.
a^(1/n) = ⁿ√a. A root in disguise. e.g. 27^(1/3) = 3, 16^(1/2) = 4.
Worked examples
Power with a negative exponent: 5⁻² = 1 / 5² = 1 / 25 = 0.04. The negative sign flips the base to its reciprocal before applying the power.
nth root: the cube root of 27 is written ³√27 = 27^(1/3) = 3, because 3 × 3 × 3 = 27. The square root of 2 is 2^(1/2) ≈ 1.414214.
Scientific notation: 149,600,000 (the distance from Earth to the Sun in km) is written 1.496 × 10⁸. The coefficient a is kept between 1 and 10, and b counts how many places the decimal point moved.
Frequently asked questions
What does an exponent (power) mean?
An exponent shows how many times a base number is multiplied by itself. In bⁿ, b is the base and n is the exponent. For example, 2⁴ = 2 × 2 × 2 × 2 = 16. Exponents provide a compact way to write repeated multiplication and are fundamental to algebra, scientific notation and growth calculations.
How do negative and decimal exponents work?
A negative exponent means take the reciprocal: a⁻ⁿ = 1/aⁿ, so 2⁻³ = 1/8 = 0.125. A fractional (decimal) exponent represents a root: a^(1/n) is the nth root of a, so 27^(1/3) = 3 and 16^0.5 = 4. Decimal exponents like 2^1.5 combine both ideas (2^1.5 = 2 × √2 ≈ 2.828).
What is the nth root of a number?
The nth root of x is the value that, raised to the power n, gives x. It is written ⁿ√x = x^(1/n). The square root (n = 2) of 25 is 5; the cube root (n = 3) of 64 is 4. This calculator computes nth roots using x^(1/n). Even roots of negative numbers are not real numbers.
What is scientific notation?
Scientific notation writes a number as a × 10ᵇ, where the coefficient a is between 1 and 10 and b is an integer exponent. It is used to express very large or very small numbers compactly — for example, 0.0000056 becomes 5.6 × 10⁻⁶ and 6,500,000 becomes 6.5 × 10⁶. It is standard in science and engineering.
What is anything raised to the power zero?
Any non-zero number raised to the power 0 equals 1, for example 9⁰ = 1 and 100⁰ = 1. This follows from the quotient rule: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰, and since any number divided by itself is 1, a⁰ = 1. The expression 0⁰ is considered indeterminate.