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Summary
Exponentiation is a mathematical operation that multiplies a number (the base) by itself a specified number of times (the exponent). Written as x^n, it means “x multiplied by itself n times.” Exponents are foundational to algebra, physics, computer science, and finance — they describe everything from compound interest to signal decay to the growth of computing power. This calculator computes any base raised to any power, including negative and fractional exponents, and expresses the result in both standard and scientific notation.
How it works
The calculator evaluates x^n for any real-valued base x and exponent n:
- Positive integer exponents — straightforward repeated multiplication. 2^10 = 2 * 2 * 2 * … (10 times) = 1024.
- Zero exponent — any non-zero number raised to the power of 0 equals 1. This follows from the exponent rule x^n / x^n = x^(n-n) = x^0 = 1.
- Negative exponents — represent reciprocals. x^(-n) = 1/x^n. So 5^(-3) = 1/5^3 = 1/125 = 0.008.
- Fractional exponents — represent roots. x^(1/n) is the nth root of x, and x^(m/n) is the nth root of x^m. For example, 8^(2/3) = (cube root of 8)^2 = 2^2 = 4.
For very large or very small results, the calculator automatically converts to scientific notation: a number between 1 and 10 multiplied by a power of 10 (e.g., 1.024 * 10^3).
Key exponent rules
| Rule | Formula | Example |
|---|---|---|
| Product | x^a * x^b = x^(a+b) | 2^3 * 2^4 = 2^7 = 128 |
| Quotient | x^a / x^b = x^(a-b) | 10^5 / 10^2 = 10^3 = 1000 |
| Power of a power | (x^a)^b = x^(ab) | (2^3)^4 = 2^12 = 4096 |
| Zero exponent | x^0 = 1 | 7^0 = 1 |
| Negative exponent | x^(-n) = 1/x^n | 3^(-2) = 1/9 |
The formulas
Where
Where
Where
Worked examples
Calculate 2^10
Repeated multiplication
= 1024
Express in scientific notation
= 1.024 * 10^3
Result
2^10 = 1024 = 1.024 * 10^3
Calculate 5^(-3)
Apply the negative exponent rule
= 1 / 5^3
Calculate 5^3
= 125
Compute the reciprocal
= 0.008
Express in scientific notation
= 8.0 * 10^(-3)
Result
5^(-3) = 0.008 = 8.0 * 10^(-3)
Express 1024 in scientific notation
Move the decimal point left until coefficient is between 1 and 10
= 1.024
Count the number of places moved to determine the exponent
= 10^3
Combine coefficient and power of 10
= 1.024 * 10^3
Result
1024 = 1.024 * 10^3
Inputs explained
- Base (x) — the number to be raised to a power. Can be any real number, including negative numbers and decimals. Note that negative bases with fractional exponents may produce complex (imaginary) results, which are outside the scope of this calculator.
- Exponent (n) — the power to raise the base to. Can be positive, negative, zero, or fractional. Fractional exponents compute roots (e.g., 0.5 for square root, 0.333… for cube root).
Outputs explained
- Result (standard form) — the computed value of x^n displayed as a regular number. For very large or very small results, this may show many digits.
- Result (scientific notation) — the same value expressed as a * 10^n, making it easier to read extreme values. For example, 2^50 = 1.1259 * 10^15 is more readable than 1,125,899,906,842,624.
- Number of digits — how many digits the result contains, useful for understanding the magnitude.
- Reciprocal — the value of x^(-n), shown alongside x^n for quick reference.
Assumptions & limitations
- Real numbers only — the calculator handles real-valued inputs and outputs. Complex results (such as (-1)^0.5 = i) are not supported; the calculator will display an error for these cases.
- Floating-point precision — JavaScript uses 64-bit floating-point arithmetic, which means results are accurate to about 15-17 significant digits. For x^n where the true result exceeds 2^53 (approximately 9 * 10^15), integer precision is lost.
- Very large exponents — exponents producing results beyond approximately 10^308 will return Infinity. Similarly, results smaller than approximately 10^(-308) will underflow to zero.
- 0^0 convention — mathematically, 0^0 is indeterminate. By convention in combinatorics and computer science, this calculator treats 0^0 as 1, consistent with JavaScript’s Math.pow(0, 0) behavior.
- Fractional exponents of negative bases — expressions like (-8)^(1/3) = -2 are mathematically valid for odd roots, but floating-point computation may return NaN. The calculator handles common cases but may not cover all edge cases.