Exponent Calculator
Calculate x raised to the power of n - including negative and fractional exponents.
What is Exponentiation?
Exponentiation is the mathematical operation of raising a number (the base) to a power (the exponent). Written as xⁿ, it means multiply x by itself n times. For example, 3⁴ = 3 × 3 × 3 × 3 = 81. Exponentiation is one of the five fundamental arithmetic operations alongside addition, subtraction, multiplication, and division.
The concept of exponents is essential across virtually every branch of science and mathematics. In physics, the inverse square law (gravity, light intensity, electric force) all follow power relationships. In finance, compound interest is expressed as P(1 + r)^n. In computer science, algorithm complexity is measured in powers: O(n²) for bubble sort, O(2^n) for brute-force combinatorics. In biology, population growth follows exponential patterns.
Negative exponents represent reciprocals: x^(−n) = 1/xⁿ. This is how very small numbers are expressed in scientific notation. The mass of an electron is approximately 9.11 × 10^(−31) kg. The Planck constant is 6.626 × 10^(−34) J·s. Without negative exponents, working with these numbers would be impractical.
Fractional exponents are another powerful generalization. x^(1/2) means the square root of x, x^(1/3) means the cube root, and in general x^(m/n) means the nth root of x^m. This connects exponentiation directly to roots, showing they are two sides of the same operation. The rules of exponents - product rule, quotient rule, power rule - all follow from the basic definition and apply to all real-number exponents.
Scientific notation expresses numbers as a × 10^b, making it practical to write both astronomically large numbers (distance to the nearest star: ~4.07 × 10^16 metres) and subatomically small ones. This calculator displays results in scientific notation when the value is very large or very small.
Formulas
How to Use This Calculator
Steps to Calculate an Exponent
Example Calculations
Example 1 - Compound Interest
₹10,000 invested at 8% per annum compounded annually for 20 years
Example 2 - Negative Exponent
Example 3 - Fractional Exponent
Frequently Asked Questions
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What is an exponent?
An exponent (or power) tells you how many times to multiply a number (the base) by itself. For example, 2^5 = 2 × 2 × 2 × 2 × 2 = 32. The base is 2 and the exponent is 5. Exponents are used throughout science, engineering, finance (compound interest), and computing.
What does a negative exponent mean?
A negative exponent means take the reciprocal. x^(−n) = 1/xⁿ. For example, 5^(−2) = 1/5² = 1/25 = 0.04. Negative exponents appear naturally in scientific notation for very small numbers: 0.001 = 10^(−3).
What does a fractional exponent mean?
A fractional exponent represents a root. x^(1/n) is the nth root of x. More generally, x^(m/n) = (ⁿ√x)^m or equivalently ⁿ√(x^m). For example, 8^(2/3) = (∛8)² = 2² = 4.
What is x to the power of 0?
Any non-zero number raised to the power 0 equals 1: x⁰ = 1. This follows from the division rule of exponents: xⁿ/xⁿ = x^(n−n) = x⁰ = 1. The expression 0⁰ is mathematically indeterminate, though in many contexts it is defined as 1 for convenience.
How do you write large numbers in scientific notation?
Scientific notation expresses a number as a × 10^b, where 1 ≤ a < 10. For example, 299,792,458 (speed of light in m/s) = 2.99792458 × 10⁸. This calculator shows results in scientific notation alongside the standard value for very large or very small numbers.
What are the laws of exponents?
The six main exponent laws: (1) a^m x a^n = a^(m+n) - multiply same base: add exponents. (2) a^m / a^n = a^(m-n) - divide same base: subtract exponents. (3) (a^m)^n = a^(mn) - power of a power: multiply exponents. (4) (ab)^n = a^n x b^n - power of a product. (5) a^0 = 1 for any a not equal to 0. (6) a^(-n) = 1/a^n. These rules apply to all real exponents.
How do you multiply two numbers with exponents?
When multiplying same bases, add exponents: a^m x a^n = a^(m+n). Example: 2^3 x 2^4 = 2^7 = 128. When dividing, subtract exponents: a^m / a^n = a^(m-n). When raising a power to a power, multiply exponents: (a^m)^n = a^(mn). These are the fundamental laws of exponents used in algebra and scientific notation.
What is the difference between 2^3 and 3^2?
2^3 = 2 x 2 x 2 = 8. 3^2 = 3 x 3 = 9. The base and exponent are not interchangeable - the result is different. In general, a^b does not equal b^a. The only exceptions are when a = b (e.g. 2^2 = 2^2), or special cases like 2^4 = 4^2 = 16. This asymmetry is important to remember when entering values in a calculator.