Exponent Calculator
Compute powers, roots, and logarithmic expressions with any base and exponent.
Results
What is an Exponent Calculator?
An exponent calculator evaluates expressions of the form x^y — a base x multiplied by itself y times — along with the two inverse operations that undo a power: roots and logarithms. Instead of grinding through repeated multiplication by hand, you enter a base and a second value, choose Power, Root, or Logarithm, and the tool returns the result, the exact formula it applied, and a short explanation of what happened.
The calculator handles negative exponents (which produce reciprocals), fractional exponents (which produce roots), and arbitrary logarithm bases, so it covers the exponent problems that come up in algebra, finance, computer science, and physics. Powers appear in everyday places more often than people expect: a 5% annual growth rate compounded for 10 years is the single expression 1.05^10 = 1.628895, meaning $1,000 grows to about $1,628.89, and one kilobyte is 2^10 = 1,024 bytes rather than an even thousand.
The Exponent Formula
All three operations in this tool rest on one relationship: if x^y = z, then the y-th root of z is x, and the logarithm of z to base x is y. Each mode applies a specific rule:
Power: x^y = x multiplied by itself y times
Root: y√x = x^(1/y)
Logarithm: log_y(x) = ln(x) ÷ ln(y)
Negative exponent: x^(-n) = 1 ÷ x^n
- Base (x) — the number being raised to a power, the value under the radical, or the argument of the logarithm.
- Exponent / Root / Log (y) — the exponent for a power, the degree for a root (2 = square root, 3 = cube root), or the base of the logarithm.
- Operation — Power (x^y), Root (y√x), or Logarithm (log_y x).
Worked Example: 2^10 = 1,024 and Its Inverse Operations
Take base 2 with exponent 10. Multiplying 2 by itself ten times runs 2, 4, 8, 16, 32, 64, 128, 256, 512, 1,024. A faster route uses the power rule: since 2^5 = 32, then 2^10 = 32 × 32 = 1,024.
The two inverse operations undo this result, which makes them a good way to check your work:
- Root: the 10th root of 1,024 asks "what number raised to the 10th power equals 1,024?" Since 1024^(1/10) = 2, the answer is 2.
- Logarithm: log base 2 of 1,024 asks "2 raised to what power equals 1,024?" By the change of base formula, ln(1024) ÷ ln(2) = 6.9314718 ÷ 0.6931472 = 10.
All three statements — 2^10 = 1,024, 10√1024 = 2, and log₂(1024) = 10 — describe the same relationship from three directions.
Common Powers Quick Reference
Use this table to check homework answers quickly or to sanity-check the calculator's output.
| Base | Squared (x²) | Cubed (x³) | To the 5th (x⁵) | To the 10th (x¹⁰) |
|---|---|---|---|---|
| 2 | 4 | 8 | 32 | 1,024 |
| 3 | 9 | 27 | 243 | 59,049 |
| 4 | 16 | 64 | 1,024 | 1,048,576 |
| 5 | 25 | 125 | 3,125 | 9,765,625 |
| 10 | 100 | 1,000 | 100,000 | 10,000,000,000 |
How to Use
- Enter the Base (x) — the number to raise to a power, take a root of, or take the logarithm of.
- Enter the Exponent / Root / Log (y) value — the exponent, the root degree, or the logarithm base, depending on the operation.
- Choose an Operation: Power (x^y), Root (y√x), or Logarithm (log_y x).
- Click Calculate to see the result, the formula used, and a short explanation. Click Reset to clear both inputs and start over.
Frequently Asked Questions
What is a negative exponent?
A negative exponent means you take the reciprocal: x^(-n) = 1/x^n. For example, 2^(-3) = 1 ÷ 2^3 = 1/8 = 0.125. When you enter a negative exponent, the calculator notes "Negative exponent: reciprocal taken" in its explanation.
How do fractional exponents work?
A fractional exponent x^(1/n) represents the n-th root of x, and x^(m/n) means the n-th root of x raised to the m-th power. For example, 8^(2/3) = (∛8)^2 = 2^2 = 4, and 27^(1/3) = 3 because 3^3 = 27.
Can I calculate any root?
Yes. Select "Root" and enter the degree as the second value — 2 for a square root, 3 for a cube root, and so on. The degree cannot be zero, and an even root of a negative number is not a real number, so the calculator shows an alert in either of those cases instead of a result.
What base can I use for logarithms?
Any positive base except 1. Common choices are 10 (common log), 2 (binary log, used in computing), and e ≈ 2.7182818 (natural log — the Log Calculator on this site handles that case directly). Base 1 is excluded because 1 raised to any power is always 1, which makes log base 1 undefined.
What is any number to the power of zero?
Any nonzero base raised to the 0 power equals 1: 5^0 = 1, 100^0 = 1, and (-3)^0 = 1. The expression 0^0 is a special case that mathematicians treat as indeterminate, but most calculators, including this one, return 1 by convention.
How does the calculator compute logarithms?
It applies the change of base formula: log_y(x) = ln(x) ÷ ln(y). For example, log_10(1000) = ln(1000) ÷ ln(10) = 6.9077553 ÷ 2.3025851 = 3. Because every logarithm is converted to natural logs first, base-2, base-10, and unusual bases like 7 all use the same method.