Exponent Calculator
Calculate any base raised to any power — positive, negative, zero, or decimal exponents. Shows the result in standard form, scientific notation, and logarithm form with step-by-step expansion.
Last updated: March 2026
How to Use the Exponent Calculator
- Enter the base — the number being multiplied. Can be positive, negative, or decimal.
- Enter the exponent — the power to raise the base to. Zero, negative, and decimal exponents are all supported.
- Read all results instantly — the main result, scientific notation, logarithm form, and step-by-step expansion appear automatically.
- Use expansion for checking — for whole number exponents up to 10, the calculator shows the multiplication expanded out (e.g., 2⁴ = 2×2×2×2 = 16) so you can verify the logic.
What Is an Exponent?
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, 3^4 = 3 × 3 × 3 × 3 = 81. We read this as "3 to the power of 4" or "3 to the 4th."
Exponents are everywhere: compound interest (A = P(1+r)^n), scientific notation (the speed of light ≈ 3 × 10^8 m/s), areas and volumes (area of a circle = πr², volume of a sphere = (4/3)πr³), and computing (storage = 2^n bytes). Mastering exponents is essential for algebra, calculus, physics, chemistry, and finance.
Positive Exponents
b^n with positive integer n means multiply b by itself n times. 5^3 = 5 × 5 × 5 = 125. The result grows rapidly — 2^10 = 1024, 2^20 = 1,048,576, 2^30 ≈ 1 billion. This exponential growth underpins Moore's Law (computing power doubles roughly every 2 years) and compound interest.
The Zero Exponent Rule
Any non-zero number raised to the power of 0 equals 1: b^0 = 1 for all b ≠ 0. Why? Using the quotient rule: b^n ÷ b^n = b^(n-n) = b^0. But any number divided by itself is 1. So b^0 = 1. The case 0^0 is indeterminate — the two rules conflict at that point.
Negative Exponents
A negative exponent means the reciprocal: b^(-n) = 1 / b^n. This follows from the quotient rule: b^0 ÷ b^n = b^(0-n) = b^(-n) = 1/b^n. So 2^(-3) = 1/2^3 = 1/8 = 0.125. Critically: negative exponents do not make results negative — they make them fractions.
Fractional (Decimal) Exponents
Fractional exponents represent roots. b^(1/2) = √b, b^(1/3) = ∛b, b^(1/n) = nth root of b. More generally: b^(m/n) = (b^(1/n))^m = (nth root of b)^m. For example: 27^(2/3) = (∛27)^2 = 3^2 = 9. This bridges the gap between powers and roots, unifying them under a single notation.
The Laws of Exponents
Six fundamental rules govern all exponent calculations. Knowing them lets you simplify complex expressions without a calculator:
1. Product Rule: b^m × b^n = b^(m+n)
When multiplying same bases, add exponents. 2^3 × 2^4 = 2^7 = 128. This is why log tables and slide rules could perform multiplication using addition.
2. Quotient Rule: b^m ÷ b^n = b^(m-n)
When dividing same bases, subtract exponents. 3^5 ÷ 3^2 = 3^3 = 27. Check: 243 ÷ 9 = 27. ✓
3. Power Rule: (b^m)^n = b^(m×n)
When raising a power to a power, multiply the exponents. (2^3)^4 = 2^12 = 4096. Check: 8^4 = 4096. ✓
4. Product to a Power: (a×b)^n = a^n × b^n
Exponents distribute over multiplication. (2×3)^4 = 6^4 = 1296, and 2^4 × 3^4 = 16 × 81 = 1296. ✓
5. Negative Exponent Rule: b^(-n) = 1/b^n
A negative exponent inverts the fraction. (3/4)^(-2) = (4/3)^2 = 16/9 ≈ 1.778.
6. Zero Exponent Rule: b^0 = 1 (for b ≠ 0)
Any non-zero base to the zero power equals 1. 1000^0 = 1. (-π)^0 = 1. 0.001^0 = 1.
Scientific Notation and Powers of 10
Scientific notation expresses any number as a × 10^n where 1 ≤ |a| < 10. It's the standard way to write very large or very small numbers in science and engineering:
Large Numbers
- Speed of light: 300,000,000 m/s = 3 × 10^8 m/s
- Distance to the Sun: ~150,000,000 km = 1.5 × 10^8 km
- Avogadro's number: 602,200,000,000,000,000,000,000 = 6.022 × 10^23
- US national debt: ~$35,000,000,000,000 = 3.5 × 10^13
Small Numbers
- Diameter of a hydrogen atom: 0.0000000001 m = 1 × 10^(-10) m
- Planck's constant: 6.626 × 10^(-34) J·s
- Proton mass: 1.67 × 10^(-27) kg
SI Prefixes and Powers of 10
- 10^3 = kilo (k) — kilometer, kilogram
- 10^6 = mega (M) — megabyte, megawatt
- 10^9 = giga (G) — gigabyte, gigahertz
- 10^12 = tera (T) — terabyte, terawatt
- 10^(-3) = milli (m) — millimeter, millisecond
- 10^(-6) = micro (μ) — micrometer, microsecond
- 10^(-9) = nano (n) — nanometer, nanosecond
Common Exponent Calculation Examples
What is 2^10?
2^10 = 1,024. This is the number of bytes in one kilobyte (2^10 = 1024) and one of the most common values in computing. 2^1=2, 2^2=4, 2^3=8, 2^4=16, 2^5=32, 2^6=64, 2^7=128, 2^8=256, 2^9=512, 2^10=1024.
What is 10^6?
10^6 = 1,000,000 (one million). Powers of 10 equal 1 followed by that many zeros. 10^6 = 1,000,000. Used in the SI prefix "mega" (e.g., 1 megabyte = 10^6 bytes in decimal convention).
What is 3^0?
3^0 = 1. Any non-zero number to the zero power equals 1. This is true for all non-zero bases: 100^0 = 1, (-5)^0 = 1, (0.001)^0 = 1.
What is 2^(-3)?
2^(-3) = 0.125 (or 1/8). Negative exponent = reciprocal: 2^(-3) = 1/2^3 = 1/8 = 0.125.
What is 4^(0.5)?
4^0.5 = 2 exactly. A 0.5 exponent = square root: 4^(1/2) = √4 = 2. Similarly, 9^0.5 = 3, 25^0.5 = 5.
What is 8^(1/3)?
8^(1/3) = 2 exactly. A 1/3 exponent = cube root: 8^(1/3) = ∛8 = 2 (since 2³ = 8).
What is (-2)^3?
(-2)^3 = -8. A negative base with an odd exponent gives a negative result: -2 × -2 × -2 = 4 × -2 = -8.
What is (-2)^4?
(-2)^4 = 16. A negative base with an even exponent gives a positive result: -2 × -2 × -2 × -2 = 4 × 4 = 16.
What is 5^(-2)?
5^(-2) = 0.04 (or 1/25). 5^(-2) = 1/5^2 = 1/25 = 0.04.
What is 1.5^4?
1.5^4 = 5.0625. 1.5 × 1.5 = 2.25, then 2.25 × 2.25 = 5.0625. Decimal bases work just like integer bases.
People Also Search For
How to calculate exponents without a calculator
For integer exponents, multiply the base by itself the specified number of times. Use the laws of exponents to break large values into manageable parts: 2^20 = (2^10)^2 = 1024^2 = 1,048,576. For fractional exponents, rewrite as roots: 64^(2/3) = (∛64)^2 = 4^2 = 16.
How to type exponents on a keyboard
In most word processors: use the superscript feature (Ctrl+Shift+= in Word). In programming: use the ^ operator (Python, many languages), ** in Python specifically, or Math.pow(base, exp) in JavaScript. In URLs and plain text, write it as 2^10 or 2**10.
What is exponential growth vs linear growth?
Linear growth adds a fixed amount each period; exponential growth multiplies by a fixed factor. Linear: 100, 200, 300, 400 (+100 each step). Exponential: 100, 200, 400, 800 (×2 each step). After 10 steps, linear reaches 1,100; exponential reaches 100 × 2^10 = 102,400. Exponential growth eventually dominates any linear growth by an enormous margin.
How are exponents used in compound interest?
Compound interest formula: A = P(1 + r/n)^(nt). Where P = principal, r = annual rate, n = compounds per year, t = years. $1,000 at 5% annual interest compounded monthly for 10 years: A = 1000 × (1 + 0.05/12)^(12×10) = 1000 × 1.004167^120 ≈ $1,647. The exponent (120) drives the compounding effect.
What is the difference between exponent and power?
In common use they mean the same thing. In the expression b^n, both n and "the power b is raised to" refer to n. Technically "power" can refer to the entire expression b^n (the result), while "exponent" refers specifically to n (the superscript). In practice, "2 to the power of 3" and "2 with an exponent of 3" both mean 2^3 = 8.
What is Euler's number e and why does e^x matter?
Euler's number e ≈ 2.71828 is the base of natural logarithms. e^x is the unique function that is its own derivative — it models continuous exponential growth and decay. Population growth, radioactive decay, compound interest (continuous compounding), and the normal distribution all involve e^x. The result of e^1 ≈ 2.718, e^2 ≈ 7.389, e^10 ≈ 22,026.
Exponent Calculator — Frequently Asked Questions
What is 2^10?
2^10 = 1,024. It equals 2 multiplied by itself 10 times. This is a fundamental value in computing — one kibibyte (KiB) = 1024 bytes = 2^10 bytes.
What does a negative exponent mean?
A negative exponent means the reciprocal: b^(-n) = 1 / b^n. For example, 10^(-3) = 1/1000 = 0.001. Negative exponents do NOT produce negative numbers — they produce fractions smaller than 1 (for bases greater than 1).
What is anything to the power of zero?
Any non-zero base to the power of 0 equals 1: b^0 = 1. This follows from the quotient rule: b^n ÷ b^n = b^0 = 1. The exception: 0^0 is mathematically indeterminate (conventionally treated as 1 in many contexts).
What is a fractional exponent?
Fractional exponents represent roots: b^(1/2) = √b (square root), b^(1/3) = ∛b (cube root), b^(m/n) = (nth root of b)^m. So 27^(2/3) = (∛27)^2 = 3^2 = 9. This unifies root and power notation under a single system.
What are the laws of exponents?
Six key rules: (1) b^m × b^n = b^(m+n) — product rule. (2) b^m ÷ b^n = b^(m-n) — quotient rule. (3) (b^m)^n = b^(mn) — power rule. (4) (ab)^n = a^n × b^n — product to power. (5) b^(-n) = 1/b^n — negative exponent. (6) b^0 = 1 — zero exponent.
What is scientific notation?
Scientific notation writes numbers as a coefficient (1 to 10) times a power of 10: 1,500,000 = 1.5 × 10^6; 0.00045 = 4.5 × 10^(-4). This calculator displays results in scientific notation when the value is very large or very small.
What is 0^0 (zero to the zero)?
0^0 is an indeterminate form — two rules conflict: b^0 = 1 suggests 0^0 = 1, while 0^n = 0 suggests 0^0 = 0. In combinatorics and many programming languages, 0^0 is defined as 1. In calculus, the limit depends on how you approach it. This calculator returns 1 for 0^0 following the most common convention.
What is the logarithm form of an exponent?
If b^n = result, then log_b(result) = n. Logarithms are the inverse of exponents. For 2^8 = 256: log_2(256) = 8. This calculator shows the log form so you can see the relationship between exponents and logarithms for each calculation.
Can I use a negative base?
Yes. For integer exponents: negative base with odd exponent = negative result ((-2)^3 = -8); even exponent = positive result ((-2)^4 = 16). For non-integer exponents with a negative base, the result may be a complex number — this calculator handles integer and simple fractional cases.
Does this calculator store my data?
No. All calculations run entirely in your browser. No data is sent to any server, stored, or logged. Your inputs are completely private.
What is 10^6?
10^6 = 1,000,000 (one million). Powers of 10 directly correspond to SI metric prefixes: kilo (10^3), mega (10^6), giga (10^9), tera (10^12). Negative powers: milli (10^-3), micro (10^-6), nano (10^-9).