Exponent Calculator
Exponents compress repeated multiplication into a tiny superscript, and this calculator works in all three directions: compute a raised to the n, solve for the exponent when you know the base and the result, or solve for the base when you know the exponent and the result. It shows the full expansion for small integer powers, accepts scientific notation, and refuses to fake answers for undefined cases like 0^0. Roots and logarithms are not separate tools here: a root is the solve-for-the-base direction, and the solve-for-the-exponent direction is a logarithm in working clothes.
Scientific notation works too: type 4e2 for 400, or 2e-6 for 0.000002.
Educational tool. Results use standard floating-point arithmetic; very large powers may display in scientific notation or as overflow.
Quick Answer
a^n means multiply a by itself n times. With the pre-filled values, 2^10 = 1024, shown in full as 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2. Negative exponents flip to the reciprocal (2^-2 = 1/4 = 0.25), fractional exponents are roots (9^0.5 = 3), and anything except zero to the power 0 is 1. Run the equation backward in the other two modes: base 2 with result 1024 gives n = 10, and exponent 3 with result 27 gives a = 3.
Key takeaways
- One tool, three directions: find the result, the exponent or the base from the other two values.
- Small integer powers show their full expansion, so a homework answer can be checked line by line.
- Undefined cases get an explanation, not a made-up number: 0^0, 0 to a negative power, base 1 in solve-for-n.
- One honest limit: solve-for-a rejects negative results, even odd roots. The workaround is in the edge cases section.
- Quick Answer
- How to use the three modes
- The formulas for all three directions
- Worked examples, verified against the tool
- The exponent rules that actually matter
- Edge cases the calculator handles honestly
- What competing calculators miss
- Frequently asked questions
- Add This Calculator to Your Website
- References
How to use the three modes
Pick the mode that matches the value you are missing, enter the two values you know, and click Calculate. The result stays hidden until you do.
- Compute a^n: enter the base and the exponent. Quick chips set e, 10 or 2 as the base, and squaring, cubing, square root or reciprocal as the exponent; a chip click fills the field and runs the calculation straight away.
- Solve for n: enter the base and the result of a^n = result. Use this for questions shaped like "2 to what power is 1024?" The base must be positive and not 1, and the result must be positive.
- Solve for a: enter the exponent and the result. Use this for "what cubed is 27?" The exponent cannot be zero, and in this mode the result must not be negative.
The solve-for-n mode is a logarithm doing its everyday job. If growth problems are your usual shape, doubling times, half-lives and per-period multipliers, the exponential growth calculator wraps the same relationship in a growth model, and the exponential and logarithmic equations calculator takes over when the unknown sits inside a fuller equation rather than a single power.
The formulas for all three directions
Each mode is the same equation, a^n = result, rearranged for a different unknown. The power mode evaluates it directly, the exponent mode isolates n with logarithms, and the base mode takes the nth root.
Exponent: n = log(result) ÷ log(a)
Base: a = result^(1/n), the nth root of the result
The log formula works with any log base, which is why the steps line shows plain log: log(result) / log(a) gives the same n whether the logs are natural or base 10, a point covered in the OpenStax College Algebra treatment of logarithmic functions. The base formula is the same move as a fractional exponent: result^(1/n) is the nth root, so solving a^3 = 27 and computing 27^(1/3) are one calculation. For a dedicated root with its simplified radical form, the cube root calculator goes deeper on that single case. Scientific notation inputs ride along free: the fields accept values such as 4e2, and writing a number that way is exactly what the convert to exponential form calculator breaks down into coefficient and exponent.
Worked examples, verified against the tool
Each mode has its own worked example pre-filled above, so all three run without typing: 2 and 10 in the power mode, 2 and 1024 in solve-for-n, and 3 and 27 in solve-for-a.
| Check | Mode | Result | Steps line shows |
|---|---|---|---|
| 2^-2 | Power | 0.25 | 2^-2 = 1 ÷ (2 × 2) = 0.25 |
| 9^0.5 | Power | 3 | 9^0.5 = 3 (fractional exponent, no expansion) |
| 5^0 | Power | 1 | Any non-zero base to the power 0 equals 1. |
| (-2)^3 | Power | -8 | -2 × -2 × -2 = -8 |
| 2^13 | Power | 8192 | 2^13 = 8192 (expansions stop past 12 factors) |
| 10^12 | Power | 1.000000e+12 | Full expansion shown; result switches to scientific notation |
| 3^n = 81 | Solve for n | 4 | n = log(81) / log(3) = 4 |
| 10^n = 0.001 | Solve for n | -3 | Negative exponents come back negative, as they should |
| a^2 = 16 | Solve for a | 4 | a = sqrt(16) = 4 |
Every row was run through the calculator code itself, so the figures and the steps wording match what the tool prints, including the formatting rules: results are rounded to 12 significant digits, and anything at or beyond 1e12, or smaller than 1e-6, switches to scientific notation. A result too large to represent at all, such as 10^400, displays as overflow rather than a wrong number.
The exponent rules that actually matter
Five rules cover nearly everything this calculator gets asked. Each one is visible in the checks above, and the full set with derivations sits in the OpenStax College Algebra section on exponents and scientific notation.
| Rule | Statement | Example |
|---|---|---|
| Negative exponent | a^-n = 1 / a^n | 2^-2 = 1/4 = 0.25 |
| Zero exponent | a^0 = 1 for any non-zero a | 5^0 = 1; 0^0 stays undefined |
| Fractional exponent | a^(m/n) is the nth root of a^m | 9^0.5 = 3, the square root of 9 |
| Power of a power | (a^m)^n = a^(m × n) | Breaks a big exponent into steps |
| Same base multiplication | a^m × a^n = a^(m + n) | 2^10 × 2^3 = 2^13 = 8192 |
Three mistakes account for most wrong answers with these rules. Typing a negative base with a fractional exponent and trusting whatever prints: if the result is not a real number, no real answer exists. Splitting a^(m/n) in the wrong order: it is the nth root of a^m, as the OpenStax section on radicals and rational exponents sets out. And treating base and exponent as interchangeable: exponentiation is not commutative, since 2^3 = 8 but 3^2 = 9.
Edge cases the calculator handles honestly
Tricky inputs get an explanation instead of a silent guess. Each message below is the exact wording the tool prints.
- 0^0: "0 raised to the power 0 is undefined." Mathematicians genuinely disagree on a convention here, so the tool makes no silent choice. One popular competitor defines it as 1 to match JavaScript; this page keeps the school and analysis convention and tells you.
- 0 to a negative power: "Zero raised to a negative power is undefined (division by zero)." A negative exponent means a reciprocal, and 1/0 has no value.
- Negative base, fractional exponent: "Not a real number: a negative base needs an integer exponent." (-8)^0.5 has no real answer; its roots live in complex numbers, which this calculator does not attempt.
- Solve for n with base 1: "Base 1 raised to any power is 1; the exponent cannot be determined." Dividing by log(1) would be dividing by zero.
- Solve for a with a negative result: the one real limitation. The mode computes result^(1/n) directly, and standard floating-point power returns not-a-number for a negative base even when n is an odd integer with a genuine real root. So a^3 = -27 gets "Not a real number: a negative result needs an odd integer root." even though the cube root of -27 is -3. The workaround: solve for the positive value (a^3 = 27 gives 3) and apply the sign yourself for odd roots. Even roots of negatives have no real answer at all, and are rejected correctly.
What competing calculators miss
The top tools for this keyword each cover part of the job well:
- AJDesigner: the closest rival, with the same three directions and arbitrary-precision integer arithmetic, a genuine edge for exact giant integers. It defines 0^0 as 1 by convention where this page reports it undefined, and it shows no repeated-multiplication expansion.
- OnlyCalculators: three unknowns plus a reciprocal readout and a powers-of-2 table. Useful extras, but the reverse solves sit behind the same form as the power calculation, and undefined cases are less explicit.
- CalculatorSoup: splits the job across separate pages, one that solves for the exponent with a clean log derivation, another for fraction exponents entered as numerator over denominator. Each page does its slice well; neither does all three directions in one place.
- Awajis: the broadest scope, with separate power, root, e^x and log operations. Breadth is its trade: each operation is shallower, without the expansion steps or the per-case error wording here.
- The gap this page fills: all three directions in one tool, expansion steps for small integer powers, quick chips for e, squares, cubes, roots and reciprocals, and error messages that name the exact reason an input has no real answer, including the negative-root limitation stated openly instead of hidden.
Frequently asked questions
An exponent tells you how many times to multiply the base by itself. In 2^3, the base 2 is multiplied by itself 3 times: 2 x 2 x 2 = 8.
Take the reciprocal: a^-n = 1 / a^n. So 2^-2 = 1 / 4 = 0.25.
A root combined with a power. a^(1/2) is the square root of a, a^(1/3) the cube root. 9^0.5 = 3.
Undefined in standard arithmetic; this calculator reports it as undefined rather than picking a convention silently.
Yes. Type 4e2 for 400 or 2e-6 for 0.000002; the input fields accept them directly.
Use the Solve for n mode: n = log(result) / log(base). For example, 2^n = 1024 gives n = 10.
Use the Solve for a mode, which computes the nth root of the result. a^3 = 27 gives a = 3.
Because the result is not a real number. Roots of negative numbers with non-integer exponents live in complex numbers, which this calculator does not attempt.
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References
- OpenStax, Rice University. College Algebra 2e: Exponents and Scientific Notation. Accessed October 21, 2026.
- OpenStax, Rice University. College Algebra 2e: Radicals and Rational Exponents. Accessed October 21, 2026.
- OpenStax, Rice University. College Algebra 2e: Logarithmic Functions. Accessed October 21, 2026.
- National Institute of Standards and Technology. Digital Library of Mathematical Functions, Section 4.2: Definitions of logarithms and powers. Accessed October 21, 2026.
- Wolfram Research. MathWorld: Exponent. Accessed October 21, 2026.
- National Institute of Standards and Technology. Digital Library of Mathematical Functions, Section 4.6: Power Series. Accessed October 21, 2026.
- Encyclopaedia Britannica. Logarithm. Accessed October 21, 2026.