Power Calculator
Base to the exponent (aⁿ) – with negative and fractional exponents
- Updated
2¹⁰
1,024
1,0242 to the power of 10 is 1,024.
How this was calculated
A whole-number exponent n tells you how many times the base is multiplied by itself. A negative exponent means the reciprocal, a fractional one a root (a^½ = √a).
RZ-26 · headon.pro
Exponentiation – basics and rules
From integer powers to negative exponents
Exponentiation is one of the fundamental arithmetic operations in mathematics. The power aⁿ means: multiply the base a by itself exactly n times. The result grows exponentially: 2¹⁰ = 1024, already over a thousand, even though the base is only 2. This explosive growth explains many natural phenomena such as compound interest, population growth, and the runtime of algorithms in computer science.
Negative exponents represent the reciprocal: a⁻ⁿ = 1/aⁿ. This allows very small numbers to be expressed elegantly. In scientific notation, 0.001 is written as 10⁻³, or the speed of light as 3 × 10⁸ m/s. Negative exponents are common in physics, chemistry, and engineering. Our calculator computes 2⁻¹ = 0.5 just as correctly as 10⁻⁶ = 0.000001.
Fractional exponents connect powers and roots: a^(1/2) = √a and a^(1/3) = ³√a. In general, a^(p/q) is the q-th root of aᵖ. This relationship is important for simplifying algebraic expressions and appears in calculus, probability theory, and physics. One key rule: a⁰ = 1 for all a ≠ 0. This also holds for seemingly tricky cases like 1000⁰ = 1.
Powers appear everywhere in everyday life: file sizes are measured in kilobytes (1 KB = 10³ bytes), megabytes (10⁶), and gigabytes (10⁹). With compound interest, a capital K grows at rate r after n years to K × (1 + r/100)ⁿ. Even at just 3% annual interest, a capital doubles after 24 years – a direct application of exponentiation that our calculator evaluates precisely.
Calculation examples
- Base
- 2
- Exponent
- 10
- Formula
- 2 × 2 × … × 2 (10 times)
- Result
- 1,024
- Base
- 2
- Exponent
- −1
- Formula
- 1 / 2¹
- Result
- 0.5
Frequently asked questions about the power calculator
Formulas, rules and examples for exponentiation
What is a power and how do you calculate it?
A power aⁿ means: multiply the base a by itself exactly n times. For example: 2³ = 2 × 2 × 2 = 8. The base is the number being multiplied, and the exponent indicates how many times. Special cases: a⁰ = 1 for any number a ≠ 0, and a¹ = a. Our power calculator computes aⁿ for any real numbers.
What do negative exponents mean?
A negative exponent means the reciprocal: a⁻ⁿ = 1/aⁿ. For example: 2⁻¹ = 1/2 = 0.5 and 10⁻³ = 1/1000 = 0.001. Negative exponents commonly appear in physics (e.g. units like m⁻¹ for "per meter"), in scientific notation, and in interest calculations. Important: the base must not be 0 if the exponent is negative.
What are fractional exponents?
Fractional exponents combine exponentiation and root extraction: a^(p/q) is the q-th root of aᵖ. For example: 8^(1/3) = ³√8 = 2 and 4^(0.5) = √4 = 2. For a negative base with a fractional exponent, there is usually no defined value within the real numbers – our calculator shows an error message in this case.
What is the difference between exponentiation and multiplication?
Multiplication (a × n) adds a n times: 3 × 4 = 3 + 3 + 3 + 3 = 12. Exponentiation (aⁿ) multiplies a by itself n times: 3⁴ = 3 × 3 × 3 × 3 = 81. Exponential growth is therefore much faster than linear growth. This explains why interest or virus infections can develop exponentially: at 7% interest per year, a capital doubles every 10 years (rule of 70).
What are the laws of exponents?
The most important laws of exponents: (1) aᵐ × aⁿ = aᵐ⁺ⁿ – same base, add the exponents. (2) aᵐ ÷ aⁿ = aᵐ⁻ⁿ – same base, subtract the exponents. (3) (aᵐ)ⁿ = aᵐⁿ – power of a power, multiply the exponents. (4) (a × b)ⁿ = aⁿ × bⁿ – power of a product. These rules greatly simplify working with expressions.