Arithmetic • Powers & Roots
Exponent Calculator
Calculate powers, square roots, cube roots, and nth roots. Supports negative and fractional exponents.
What This Exponent Calculator Does
This exponent calculator evaluates expressions of the form xⁿ and ⁿ√x for any real numbers. It supports positive exponents, negative exponents (which produce reciprocals), fractional exponents (which produce roots), and zero exponents. The calculator also handles square roots (√), cube roots (³√), and arbitrary nth roots, making it a complete tool for all exponentiation and root operations.
How Exponents Work
Exponentiation is repeated multiplication: xⁿ means x multiplied by itself n times. For example, 2³ = 2 × 2 × 2 = 8. But exponents can be more than just whole numbers:
- Positive exponent: xⁿ means x multiplied by itself n times. Example: 5³ = 125
- Negative exponent: x⁻ⁿ = 1/xⁿ. Example: 2⁻³ = 1/8 = 0.125
- Zero exponent: x⁰ = 1 (for any x ≠ 0). Example: 7⁰ = 1
- Fractional exponent: x^(1/n) = ⁿ√x. Example: 27^(1/3) = ³√27 = 3
- General fractional exponent: x^(m/n) = (ⁿ√x)ᵐ = ⁿ√(xᵐ). Example: 8^(2/3) = (³√8)² = 2² = 4
How to Use This Calculator
- For powers (xⁿ): Enter the base (x) and the exponent (n). The exponent can be any real number — positive, negative, fractional, or zero.
- For roots (ⁿ√x): Enter the number (x) and the root degree (n). For square roots, enter n = 2; for cube roots, n = 3; for any other root, enter the corresponding degree.
- Read the result: The calculator displays the result in both numeric and symbolic form (e.g., 2³ = 8 or ³√27 = 3).
Example Calculations
- Basic power: 2³ = 2 × 2 × 2 = 8
- Negative exponent: 2⁻³ = 1/2³ = 1/8 = 0.125
- Fractional exponent (square root): 9^(1/2) = √9 = 3
- Fractional exponent (cube root): 27^(1/3) = ³√27 = 3
- Zero exponent: 5⁰ = 1
- Scientific notation: 10⁶ = 1,000,000; 10⁻⁶ = 0.000001
Common Applications
- Compound Interest: A = P(1 + r/n)^(nt) — the exponent represents the number of compounding periods.
- Scientific Notation: Numbers expressed as a × 10ⁿ, where n is an integer exponent representing order of magnitude.
- Growth and Decay: Exponential growth models use exponents to represent growth rates and time periods.
- Physics: Inverse square laws (like gravity and light intensity) use negative exponents to represent diminishing effects.
- Computer Science: Powers of 2 (2¹⁰ = 1024) are fundamental to digital storage and computing.
Frequently Asked Questions
What does a negative exponent mean?
A negative exponent means take the reciprocal of the base raised to the positive version of that exponent. For example, 2⁻³ = 1/2³ = 1/8.
What does a fractional exponent mean?
A fractional exponent represents a root. b^(1/2) is the square root of b, and more generally b^(m/n) is the n-th root of b raised to the m-th power.
Why does any number to the power of 0 equal 1?
It follows from the pattern of dividing consecutive powers of the same base (b³/b²=b, b²/b¹=b, b¹/b⁰=b) — for the pattern to hold consistently, b⁰ must equal 1 for any nonzero b.
What happens when I raise a negative number to an exponent?
A negative base raised to an even exponent gives a positive result. A negative base raised to an odd exponent gives a negative result. For fractional exponents, the result may be complex if the root index is even and the base is negative.
What is scientific notation and how does it relate to exponents?
Scientific notation expresses numbers as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer exponent. It's a compact way to write very large or very small numbers — for example, 3.0 × 10⁸ = 300,000,000.