Exponential function xy
Calculator and formula for calculating the power value with base x and exponent y
Power value calculator
What is calculated?
This function computes the power value for the given base x and exponent y. Both arguments must be real numbers. The complex Exp function is available here.
Power function info
Properties
General power function x^y:
- Base x can be chosen freely
- Exponent y can be chosen freely
- For positive base: always defined
- For negative base: only for integer exponents
- Base 0: only for positive exponents
Note: For complex power functions use the complex Exp page.
Examples
2⁰ = 1
Any number to the power of 0 equals 1
Any number to the power of 0 equals 1
2¹ = 2
Base to the power 1 equals the base
Base to the power 1 equals the base
2² = 4
Square of 2
Square of 2
2³ = 8
Cube of 2
Cube of 2
3² = 9
Different bases are possible
Different bases are possible
Formulas of the general power function
General form
\[f(x,y) = x^y\]
Power function with arbitrary base
Rewrite with e
\[x^y = e^{y \cdot \ln(x)}\]
Using natural exponential
Rewrite with base 10
\[x^y = 10^{y \cdot \log_{10}(x)}\]
Using base-10 logarithm
Power laws
\[x^a \cdot x^b = x^{a+b}\]
Multiplication of equal bases
Further power laws
\[(x^a)^b = x^{a \cdot b}\]
Power of a power
Division of powers
\[\frac{x^a}{x^b} = x^{a-b}\]
Division of equal bases
Calculation examples
Example 1: calculate 2⁴
\[2^4 = 2 \times 2 \times 2 \times 2 = 16\]
The base 2 is multiplied by itself 4 times.
Example 2: calculate 3²
\[3^2 = 3 \times 3 = 9\]
The square of 3 is 9.
Example 3: calculate 5⁰
\[5^0 = 1\]
Any number (except 0) to the power of 0 equals 1.
Example 4: calculate 10³
\[10^3 = 10 \times 10 \times 10 = 1000\]
Powers of 10 are especially straightforward.
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