CalculateY
Quick Answer
2 raised to the 8th power equals 256. 5 raised to the 3rd power equals 125.
Common Examples
| Input | Result |
|---|---|
| 2^8 | 256 |
| 5^3 | 125 |
| 10^-2 | 0.01 |
| 27^(1/3) | 3 (cube root of 27) |
| 7^0 | 1 |
How It Works
The Formula
Exponentiation is defined as repeated multiplication of a base number:
b^n = b x b x b x … x b (n times)
This definition extends naturally to several special cases:
Zero exponent: Any nonzero number raised to the power of zero equals 1. By convention, 0^0 is also defined as 1 in most contexts (combinatorics, set theory, and computer science).
b^0 = 1 (for any b)
Negative exponent: A negative exponent means the reciprocal of the positive power. This follows from the exponent rule b^m / b^n = b^(m-n). When m = 0, dividing gives b^(-n) = 1 / b^n.
b^(-n) = 1 / b^n
Fractional exponent: A fractional exponent represents a root. The denominator of the fraction specifies the root, and the numerator specifies the power. For example, b^(1/2) is the square root and b^(1/3) is the cube root.
b^(m/n) = nth root of (b^m)
Key Exponent Laws
These identities are useful for simplifying expressions with exponents:
- Product rule: b^m x b^n = b^(m+n)
- Quotient rule: b^m / b^n = b^(m-n)
- Power rule: (b^m)^n = b^(m x n)
- Product to power: (a x b)^n = a^n x b^n
Worked Example
To calculate 3^5: multiply 3 by itself 5 times. 3 x 3 = 9, 9 x 3 = 27, 27 x 3 = 81, 81 x 3 = 243. So 3^5 = 243.
To calculate 2^(-4): first compute 2^4 = 16, then take the reciprocal: 1/16 = 0.0625. So 2^(-4) = 0.0625.
To calculate 16^(3/4): first find the 4th root of 16, which is 2. Then raise 2 to the 3rd power: 2^3 = 8. So 16^(3/4) = 8.