Exponent Calculator
Raise a number to a power (bⁿ), solve for the base given a result and exponent, or find the exponent given a base and result. Negative exponents, fractional exponents, and Euler's number e as a base are all supported. Pick a mode, enter any two values, and get the third.
Formula
Worked example
2^10 = 2 * 2 * ... * 2 (ten times) = 1,024. Reverse: given b^3 = 27, the base is 27^(1/3) = 3. Exponent solve: 2^n = 1024, so n = log(1024)/log(2) = 10.
What an exponent means
An exponent tells you how many times to multiply a base by itself. In bⁿ, b is the base and n is the exponent or power. So 2^3 means 2 * 2 * 2 = 8. Exponents are a compact way to write repeated multiplication and appear everywhere from compound interest to scientific notation and computer science. This calculator handles all three unknowns: enter any two of the base, exponent, and result to find the third.
Zero, negative and fractional exponents
Three special cases extend the idea beyond whole positive numbers. Any non-zero base raised to the power 0 equals 1 (the zero-exponent rule). A negative exponent means take the reciprocal: b^(-n) = 1 / b^n, so 2^(-3) = 1/8 = 0.125. A fractional exponent is a root: b^(1/2) is the square root, b^(1/3) is the cube root, and b^(p/q) is the q-th root of b^p. These rules keep the laws of exponents consistent across all values of n.
Euler's number e as a base
Euler's number e (approximately 2.71828) is the most natural base for exponential functions. e^x describes continuous exponential growth and appears throughout calculus, probability, and physics. The natural logarithm ln(x) is the inverse: e^(ln x) = x. Toggle "Use e as base" to calculate e^n, which is useful for continuous compound interest (A = Pe^(rt)) and half-life or decay models.
Solving for the base or the exponent
This calculator works in three directions. To find the result, enter a base and exponent. To find an unknown base (for example, "what number cubed gives 125?"), switch to "Solve for base", enter the exponent and result, and the calculator computes b = result^(1/n). To find an unknown exponent (for example, "2 to what power gives 1024?"), switch to "Solve for exponent" and use the change-of-base formula: n = log(result) / log(base).
Why exponents grow so fast
Exponential growth outpaces almost everything. Each time the exponent rises by one, the result multiplies by the entire base again. So values balloon quickly: 2^10 is about 1,000, 2^20 is over 1,000,000, and 2^30 exceeds 1,000,000,000. This explosive growth is why compound interest, computer storage, and viral spread all follow exponential curves. The chart (shown for integer exponents up to 20) illustrates how steeply the curve rises.
Laws of exponents
| Rule | Formula | Example |
|---|---|---|
| Product rule | b^m * b^n = b^(m+n) | 2^3 * 2^4 = 2^7 = 128 |
| Quotient rule | b^m / b^n = b^(m-n) | 5^5 / 5^2 = 5^3 = 125 |
| Power of a power | (b^m)^n = b^(m*n) | (3^2)^3 = 3^6 = 729 |
| Power of a product | (a*b)^n = a^n * b^n | (2*3)^2 = 4 * 9 = 36 |
| Power of a quotient | (a/b)^n = a^n / b^n | (4/2)^3 = 64/8 = 8 |
| Zero exponent | b^0 = 1 (b not 0) | 7^0 = 1 |
| Negative exponent | b^(-n) = 1 / b^n | 2^(-3) = 1/8 = 0.125 |
| Fractional exponent | b^(1/n) = n-th root of b | 8^(1/3) = cube root of 8 = 2 |
| General fractional | b^(m/n) = n-th root of b^m | 8^(2/3) = cube root of 64 = 4 |
These rules hold for any real base b and exponents m and n (with b not 0 where noted).
Frequently asked questions
What is any number to the power of zero?
Any non-zero number raised to the power 0 equals 1. This follows from the quotient rule: b^n / b^n = b^(n-n) = b^0, and b^n / b^n is also 1. The case 0^0 is conventionally treated as 1 in algebra but is sometimes left undefined in analysis.
What does a negative exponent do?
A negative exponent gives the reciprocal of the positive power: b^(-n) = 1 / b^n. For example, 5^(-2) = 1 / 25 = 0.04. This keeps the quotient rule b^m / b^n = b^(m-n) consistent when m is less than n.
How do I solve for an unknown base?
If you know the result a and the exponent n, switch the mode to "Solve for base". The calculator computes b = a^(1/n), which is the n-th root of a. For example, b^3 = 27 means b = 27^(1/3) = 3.
How do I find an unknown exponent?
Switch to "Solve for exponent" and enter the base and result. The calculator applies the change-of-base formula: n = log(result) / log(base). For example, 2^n = 1024 gives n = log(1024) / log(2) = 10. The base must be positive and not equal to 1.
Why can a negative base give "not a real number"?
Raising a negative base to a non-integer exponent, such as (-4)^0.5 (the square root of -4), has no real-number answer. The result is a complex number. This calculator works in the real numbers only, so it returns "Not a real number" in those cases.
What is e and why is it special?
Euler's number e (approximately 2.71828) is the unique base for which the derivative of e^x equals e^x. It arises naturally in continuous compound interest, population models, and the normal distribution. The natural logarithm ln(x) is its inverse, making e^x and ln(x) a foundational pair in mathematics.
How do I multiply or divide numbers with the same base?
To multiply, add the exponents: b^m * b^n = b^(m+n). For example, 2^3 * 2^4 = 2^7 = 128. To divide, subtract the exponents: b^m / b^n = b^(m-n). For example, 3^5 / 3^2 = 3^3 = 27. The full set of rules is in the reference table above.