How exponents work
An exponent (or power) tells you how many times to multiply the base by itself: base^exponent. So 2^10 means 2 multiplied by itself 10 times, which equals 1,024. A negative exponent means a reciprocal (2^−2 = 1/4), and an exponent of 0 always gives 1.
The rule for zero follows from the pattern of division. Each time the exponent drops by one, the value is divided by the base: 2^3 = 8, 2^2 = 4, 2^1 = 2, and one more step gives 2^0 = 1. Continue past zero and you land on the reciprocals, which is exactly what negative exponents describe. Very large or very small powers are easier to read in the compact form produced by the scientific notation calculator.
How to use this calculator
Enter the base and the exponent. Both can be negative or decimal — a fractional exponent is a root (4^0.5 = √4 = 2). For square roots specifically, see the square root calculator.
A fraction in the exponent has two jobs at once: the denominator takes a root and the numerator raises to a power. So 8^(2/3) means the cube root of 8, squared, which is 4. Be aware that a negative base combined with a fractional exponent has no real answer, since it asks for an even root of a negative number.
The exponent rules
Three rules cover almost every manipulation you will need:
- Same base, multiplying: add the exponents, so 2^3 × 2^7 = 2^10 = 1,024.
- Same base, dividing: subtract the exponents.
- Power of a power: multiply the exponents, so (2^5)^2 = 2^10.
Exponential growth is startlingly fast, which is why powers show up in compound interest, data storage and population models. Doubling ten times multiplies a quantity by 1,024, not by 20 — the intuition most people start with is off by a factor of fifty. The operation that recovers the exponent from the result is the logarithm, handled by the logarithm calculator.
Frequently asked questions
- How do I calculate an exponent?
- Multiply the base by itself as many times as the exponent says. 2^10 = 1,024.
- What is 2 to the power of 10?
- 1,024.
- What does a negative exponent mean?
- It's a reciprocal: base^−n = 1 ÷ base^n. So 2^−2 = 1/4.
- What is anything to the power of 0?
- 1. Any nonzero number raised to the power 0 equals 1.
- Is −2^2 the same as (−2)^2?
- No. Without brackets the exponent binds tighter than the minus sign, so −2^2 means −(2^2) = −4, while (−2)^2 = 4. Always bracket a negative base when you want the sign included in the power.