Exponent Calculator – Calculate Base to the Power of n
Easily calculate any number raised to a power with our free exponent calculator. Supports positive, negative, and fractional exponents for fast and accurate results with step-by-step solutions.
Understanding Exponents
An exponent tells you how many times to multiply a number by itself. In 2³, the base is 2 and the exponent is 3, meaning 2 × 2 × 2 = 8. Exponents are a compact way to write repeated multiplication, just as multiplication is a compact way to write repeated addition.
Exponents appear everywhere in math and science. Compound interest uses exponents to calculate growth. Physics uses them for inverse-square laws (gravity, light intensity). Computer science uses powers of 2 for memory sizes. Understanding exponents is essential for algebra and beyond.
Exponent Rules
Basic Rules
Negative Exponents
Negative exponents mean "one over" the positive power.
Fractional Exponents
Fractional exponents represent roots.
Power Rules
Worked Examples
Example 1: 2¹⁰
Base: 2, Exponent: 10
Expanding: 2¹⁰ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
Calculating: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024
Result: 1024
Powers of 2 are fundamental in computing. 2¹⁰ = 1024 is approximately 1000 (kilo in binary).
Example 2: 3⁴
Base: 3, Exponent: 4
Expanding: 3⁴ = 3 × 3 × 3 × 3
Calculating: 3 × 3 = 9, 9 × 3 = 27, 27 × 3 = 81
Result: 81
Three to the fourth power. Each multiplication triples the previous result.
Example 3: 5⁻²
Base: 5, Exponent: -2
Negative exponent rule: 5⁻² = 1/5²
Calculate positive power: 5² = 25
Result: 1/25 = 0.04
Negative exponents give reciprocals. The result is less than 1.
Example 4: 10⁶
Base: 10, Exponent: 6
10⁶ = 10 × 10 × 10 × 10 × 10 × 10
Result: 1,000,000 (one million)
Powers of 10 are easy: just write 1 followed by that many zeros. 10⁶ is the metric prefix "mega".
Example 5: 2^0.5 (Square Root of 2)
Base: 2, Exponent: 0.5 (or 1/2)
Fractional exponent: 2^(1/2) = √2
Result: 1.414213...
The square root of 2 is irrational – its decimal never ends or repeats. It's approximately 1.414.
Example 6: 7⁰
Base: 7, Exponent: 0
Zero exponent rule: any non-zero number to the power 0 equals 1
Result: 1
Why is a⁰ = 1? Because aⁿ ÷ aⁿ = a⁰ = 1. It keeps the exponent rules consistent.
Example 7: 1.5³
Base: 1.5, Exponent: 3
Expanding: 1.5³ = 1.5 × 1.5 × 1.5
Calculating: 1.5 × 1.5 = 2.25, 2.25 × 1.5 = 3.375
Result: 3.375
Decimal bases work the same way. 1.5³ = (3/2)³ = 27/8 = 3.375.
Quick Fact
The word "exponent" was coined by Michael Stifel in 1544. It comes from Latin meaning "placed out" – the exponent is placed out (above) the base. Before exponents, mathematicians wrote out repeated multiplication in full. René Descartes introduced the modern notation (like x³) in 1637.
Frequently Asked Questions
What does an exponent of 0 mean?
Any non-zero number raised to the power 0 equals 1. This keeps the exponent rules consistent: aᵐ ÷ aᵐ = aᵐ⁻ᵐ = a⁰ = 1. Zero to the power 0 is undefined (or defined as 1 in some contexts).
How do negative exponents work?
A negative exponent means "one over" the positive power: a⁻ⁿ = 1/aⁿ. So 2⁻³ = 1/2³ = 1/8. Negative exponents give values less than 1 (for bases greater than 1).
What's a fractional exponent?
Fractional exponents represent roots. a^(1/2) is the square root of a. a^(1/3) is the cube root. a^(3/2) means square root of a, then cube it (or vice versa).
Why is 0⁰ undefined?
There are conflicting limits: 0ⁿ approaches 0 as n approaches 0, but n⁰ approaches 1 as n approaches 0. In some contexts (like combinatorics), 0⁰ is defined as 1 for convenience.
How do I calculate large exponents?
For very large exponents, use a calculator or computer. For mental math, break it down: 2¹⁰ = (2⁵)² = 32² = 1024. Use exponent rules to simplify before calculating.
What's the difference between -2² and (-2)²?
Order of operations matters! -2² = -(2²) = -4. But (-2)² = (-2) × (-2) = 4. Parentheses change everything. Always use parentheses for negative bases.
Other Free Tools
Square Root Calculator – Compute √ of Any Number
Find the square root of any number instantly with our free online square root calculator. Supports both perfect and imperfect squares with high precision decimal results.
Logarithm Calculator – Compute Log of Any Base Online
Calculate logarithms of any number for any base with our free online logarithm calculator. Supports log base 10, natural log (ln), and custom base logarithms with instant results.
Free Online Scientific Calculator – Advanced Math Functions
Perform complex scientific calculations online with our free scientific calculator. Supports trigonometry, logarithms, exponents, roots, and more – ideal for students and professionals.
Free Online Standard Calculator – Fast & Easy Math
Use our free standard calculator online to perform quick arithmetic operations including addition, subtraction, multiplication, and division. Simple, fast, and accurate for everyday math needs.
Scientific Notation Converter – Standard to Scientific Form
Convert any number to scientific notation or from scientific notation to standard form with our free online converter. Perfect for chemistry, physics, and large number calculations.
Taylor Series Calculator – Expand Functions as Taylor Series
Generate Taylor and Maclaurin series expansions of any function with our free online calculator. Compute series up to any number of terms with coefficient and error visualization.
Exponent Calculator – Calculate Base to the Power of n
Easily calculate any number raised to a power with our free exponent calculator. Supports positive, negative, and fractional exponents for fast and accurate results.
Derivative Calculator – Differentiate Functions Step by Step
Calculate the derivative of any function with our free online derivative calculator. Applies power, product, quotient, and chain rules with detailed step-by-step differentiation shown.