Exponent Calculator
Calculate powers and exponents, solve exponential expressions, and learn exponent rules.
π’Exponent Calculator
Calculate powers and exponents with step-by-step solutions
Exponent Rules Quick Reference
Related Calculators
About This Calculator
"What is 2 to the power of 10?" This is the kind of question our Exponent Calculator answers instantly.
Exponents are one of the most fundamental concepts in mathematics, appearing everywhere from compound interest calculations to computer science algorithms. An exponent (also called a power or index) tells you how many times to multiply a number by itself. For example, 2^3 means 2 x 2 x 2 = 8. While simple exponents are easy to calculate mentally, things get more complex with larger numbers, negative exponents, or fractional powers.
This free online exponent calculator handles all types of power calculations with ease. Enter any base number and exponent to get instant results, whether you're working with positive integers, negative exponents (which create fractions), or fractional exponents (which represent roots). The calculator provides step-by-step explanations so you can understand exactly how each result is derived.
Our tool automatically converts extremely large or small results to scientific notation, making it perfect for scientific and engineering applications. Whether you're a student learning the laws of exponents, an investor calculating compound returns, a scientist working with exponential decay, or anyone who needs fast and accurate power calculations, this calculator delivers precise results every time. Understanding exponents opens doors to advanced mathematics including logarithms, exponential growth models, and calculus.
How to Use the Exponent Calculator
- 1**Enter the base number**: Type the number you want to raise to a power in the Base (x) field. This can be any real number including decimals and negatives.
- 2**Enter the exponent**: Type the power you want to raise the base to in the Exponent (n) field. Supports whole numbers, decimals, and negative values.
- 3**Use quick presets**: Click the preset buttons (x^2, x^3, x^4, x^-1, x^0.5) for common calculations.
- 4**Toggle scientific notation**: Enable the switch to always display results in scientific notation format.
- 5**Review step-by-step solution**: The calculator shows how the result is calculated, explaining the mathematical process.
- 6**Copy your result**: Click the copy button to copy the result to your clipboard.
- 7**Save to recent calculations**: Click the save button to add this calculation to your history for future reference.
Formula
x^n = x multiplied by itself n timesThe exponent n tells you how many times to multiply the base x by itself. For example, 2^5 = 2 x 2 x 2 x 2 x 2 = 32.
Exponent Rules
Understanding the laws of exponents is essential for simplifying expressions and solving equations efficiently:
Product Rule: When multiplying powers with the same base, add the exponents.
- Formula: a^m x a^n = a^(m+n)
- Example: 2^3 x 2^4 = 2^7 = 128
Quotient Rule: When dividing powers with the same base, subtract the exponents.
- Formula: a^m / a^n = a^(m-n)
- Example: 5^6 / 5^2 = 5^4 = 625
Power of a Power Rule: When raising a power to another power, multiply the exponents.
- Formula: (a^m)^n = a^(m x n)
- Example: (2^3)^2 = 2^6 = 64
Power of a Product Rule: When raising a product to a power, raise each factor to that power.
- Formula: (ab)^n = a^n x b^n
- Example: (3 x 4)^2 = 3^2 x 4^2 = 144
Zero Exponent Rule: Any non-zero number raised to the power of zero equals 1.
- Formula: a^0 = 1 (where a is not 0)
- Example: 999^0 = 1
Negative Exponents
A negative exponent indicates the reciprocal (or inverse) of the positive exponent:
The Rule: a^(-n) = 1 / a^n
A negative exponent 'flips' the base to the denominator of a fraction.
Examples:
| Expression | Calculation | Result |
|---|---|---|
| 2^(-1) | 1/2^1 | 0.5 |
| 2^(-2) | 1/2^2 | 0.25 |
| 2^(-3) | 1/2^3 | 0.125 |
| 10^(-1) | 1/10 | 0.1 |
| 10^(-2) | 1/100 | 0.01 |
| 10^(-3) | 1/1000 | 0.001 |
Why This Makes Sense: Look at the pattern of decreasing powers: 2^3=8, 2^2=4, 2^1=2, 2^0=1, 2^(-1)=0.5. Each step divides by 2, so negative exponents naturally continue this pattern.
Applications: Negative exponents are essential for scientific notation (e.g., 3.5 x 10^(-4) = 0.00035), decay rates in physics, and expressing very small probabilities.
Fractional Exponents
Fractional exponents provide an elegant way to express roots:
The Connection: a^(1/n) = the nth root of a
Common Fractional Exponents:
| Exponent | Meaning | Example |
|---|---|---|
| a^(1/2) | Square root | 16^0.5 = 4 |
| a^(1/3) | Cube root | 27^(1/3) = 3 |
| a^(1/4) | Fourth root | 81^0.25 = 3 |
| a^(2/3) | Cube root, then square | 8^(2/3) = 4 |
| a^(3/2) | Square root, then cube | 4^1.5 = 8 |
The General Rule: a^(m/n) = (nth root of a)^m = nth root of (a^m)
Both approaches yield the same result: 8^(2/3) = (cube root of 8)^2 = 2^2 = 4
Important Notes:
- Even roots of negative numbers are not real (e.g., (-4)^0.5 produces a complex number)
- Odd roots of negative numbers are real (e.g., (-8)^(1/3) = -2)
Scientific Notation
When working with very large or very small numbers, scientific notation uses exponents to express values compactly:
Format: a x 10^n (where 1 <= a < 10)
Large Numbers:
| Standard Form | Scientific Notation | Name |
|---|---|---|
| 1,000 | 1 x 10^3 | Thousand |
| 1,000,000 | 1 x 10^6 | Million |
| 1,000,000,000 | 1 x 10^9 | Billion |
| 1,000,000,000,000 | 1 x 10^12 | Trillion |
Small Numbers:
| Standard Form | Scientific Notation |
|---|---|
| 0.001 | 1 x 10^(-3) |
| 0.000001 | 1 x 10^(-6) |
| 0.000000001 | 1 x 10^(-9) |
Powers of 2 in Computing:
| Power | Value | Usage |
|---|---|---|
| 2^10 | 1,024 | Kilobyte |
| 2^20 | 1,048,576 | Megabyte |
| 2^30 | 1,073,741,824 | Gigabyte |
| 2^64 | ~1.84 x 10^19 | 64-bit limit |
Real-World Applications
Exponents appear in virtually every scientific, financial, and technical field:
Finance and Economics:
- Compound Interest: A = P(1 + r/n)^(nt)
- Present Value: PV = FV / (1 + r)^n
- The Rule of 72: Years to double β 72 / interest rate %
Physics:
- Kinetic Energy: KE = 1/2 mv^2
- Gravitational Force: F = Gm1m2/r^2
- Einstein's equation: E = mc^2
- Radioactive Decay: N = N0 e^(-lambda t)
Biology:
- Population Growth: P = P0 e^(rt)
- Bacterial Doubling: N = N0 x 2^(t/d)
- DNA Amplification (PCR): 2^n copies after n cycles
Computer Science:
- Binary Numbers: Each bit represents 2^n
- Algorithm Complexity: O(n^2), O(2^n), O(log n)
- Memory Addressing: 2^32 addresses (32-bit), 2^64 (64-bit)
Chemistry:
- pH Scale: pH = -log[H+]
- Arrhenius Equation: k = Ae^(-Ea/RT)
Pro Tips
- π‘Remember that any non-zero number raised to the power of 0 equals 1. This is a fundamental rule: 5^0 = 1, 100^0 = 1, (-7)^0 = 1.
- π‘Negative exponents create fractions. Think of a^(-n) as '1 divided by a^n'. So 2^(-3) = 1/8 = 0.125.
- π‘Fractional exponents are roots. a^(1/2) is square root, a^(1/3) is cube root. Remember: a^(m/n) = nth root of (a^m).
- π‘When multiplying same bases, add exponents: 2^3 x 2^4 = 2^7. When dividing, subtract: 2^7 / 2^3 = 2^4.
- π‘For power of a power, multiply exponents: (2^3)^4 = 2^12 = 4096.
- π‘Memorize key powers of 2: 2^10 = 1024 (important for computing). Double it for each additional power.
- π‘Use the Rule of 72 to quickly estimate doubling time: years to double is approximately 72 divided by the interest rate percentage.
- π‘Scientific notation uses powers of 10. Move decimal right equals positive exponent, left equals negative.
- π‘Check your work: if 2^3 = 8, then 8^(1/3) should equal 2. Exponents and roots are inverses.
- π‘For compound interest, remember A = P(1+r)^n. The exponent n is the number of compounding periods.
- π‘Exponents grow faster than you might expect. 2^10 = 1,024 but 2^20 = over 1 million!
- π‘When in doubt about exponent rules, substitute small numbers to verify: test (2^2)^3 = 4^3 = 64 = 2^6.
Frequently Asked Questions
An exponent (also called a power or index) tells you how many times to multiply a number by itself. In the expression b^n, b is the base and n is the exponent. For example, 2^3 means 2 x 2 x 2 = 8. The base (2) is multiplied by itself the number of times indicated by the exponent (3).