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Calculate powers and exponents (base^exponent). Supports positive, negative, zero, and fractional exponents with scientific notation for large results.
Can be positive, negative, zero, or fractional
Enter base and exponent to calculate
Enter the base number (the number being multiplied by itself). This can be any real number.
Enter the exponent (power). This can be positive, negative, zero, or even a fraction/decimal.
Click Calculate to see the result. Large numbers are automatically converted to scientific notation for readability.
a^0 = 1
Any number to the power of 0 equals 1 (except 0^0)
a^(-n) = 1/(a^n)
Example: 2^(-3) = 1/(2^3) = 1/8 = 0.125
a^m × a^n = a^(m+n)
Example: 2^3 × 2^2 = 2^5 = 32
a^m / a^n = a^(m-n)
Example: 5^4 / 5^2 = 5^2 = 25
(a^m)^n = a^(m×n)
Example: (3^2)^3 = 3^6 = 729
a^(1/n) = ∛a (nth root)
Example: 16^(1/2) = √16 = 4
| Base | Power of 2 | Power of 3 | Power of 4 |
|---|---|---|---|
| 2 | 2² = 4 | 2³ = 8 | 2⁴ = 16 |
| 3 | 3² = 9 | 3³ = 27 | 3⁴ = 81 |
| 4 | 4² = 16 | 4³ = 64 | 4⁴ = 256 |
| 5 | 5² = 25 | 5³ = 125 | 5⁴ = 625 |
| 6 | 6² = 36 | 6³ = 216 | 6⁴ = 1296 |
| 7 | 7² = 49 | 7³ = 343 | 7⁴ = 2401 |
| 8 | 8² = 64 | 8³ = 512 | 8⁴ = 4096 |
| 9 | 9² = 81 | 9³ = 729 | 9⁴ = 6561 |
| 10 | 10² = 100 | 10³ = 1000 | 10⁴ = 10000 |
An exponent tells you how many times to multiply a number by itself. In a^n, 'a' is the base and 'n' is the exponent. For example, 2^3 means 2 × 2 × 2 = 8. The exponent is also called the power or index.
A negative exponent means you take the reciprocal and use the positive exponent. For example, 2^(-3) = 1/(2^3) = 1/8 = 0.125. It's a way to represent division instead of multiplication.
This comes from the quotient rule: a^m / a^n = a^(m-n). If m = n, then a^m / a^m = 1 (any number divided by itself is 1), and also a^(m-m) = a^0. Therefore, a^0 = 1.
A fractional exponent represents a root. The denominator indicates which root to take. For example, a^(1/2) is the square root, a^(1/3) is the cube root. More generally, a^(m/n) = ∛(a^m), the nth root of a to the mth power.
For very large exponents, calculators use logarithms and scientific notation. For example, 2^100 = 1.27 × 10^30. You can also use repeated squaring: to find 2^8, calculate 2^2 = 4, then 4^2 = 16, then 16^2 = 256.
Exponents are used in compound interest calculations, population growth, radioactive decay, computing (powers of 2), chemistry (pH calculations), physics (inverse square laws), and engineering (scaling and dimensional analysis).
Exponents are a shorthand way of expressing repeated multiplication. Instead of writing 2 × 2 × 2 × 2, we can write 2^4. This notation becomes incredibly useful when dealing with large numbers or complex calculations.
The power of exponents extends beyond simple multiplication. Negative exponents represent division, fractional exponents represent roots, and zero exponents have a special property. These rules make exponents a versatile tool in mathematics and science.