Exponent Calculator

Raise any base to any power. Supports decimals and negative numbers. Results update instantly.

Result
1,024
2^10 = 1,024

Common Powers of 2

Powers x2 through x10 using the base entered above.

Power Expression Result Scientific Notation

Logarithm Calculator

Enter a positive number to compute its logarithms.

log10(x)
3
ln(x) — natural log
6.9078
log2(x)
9.9658
10^3 = 1,000  |  e^6.9078 ≈ 1,000  |  2^9.9658 ≈ 1,000

Rules of Exponents — Reference Card

These seven rules cover every exponent manipulation you will encounter in algebra, physics, and engineering.

Product Rule
a^m × a^n = a^(m+n)
2^3 × 2^4 = 2^7 = 128
Quotient Rule
a^m ÷ a^n = a^(m-n)
2^7 ÷ 2^3 = 2^4 = 16
Power Rule
(a^m)^n = a^(m×n)
(2^3)^4 = 2^12 = 4,096
Zero Exponent
a^0 = 1 (a ≠ 0)
7^0 = 1, 999^0 = 1
Negative Exponent
a^-n = 1 / a^n
2^-3 = 1/8 = 0.125
Fractional Exponent
a^(1/n) = n-th root of a
8^(1/3) = cube root of 8 = 2
Product of Powers
(a × b)^n = a^n × b^n
(2 × 3)^2 = 4 × 9 = 36
Quotient of Powers
(a / b)^n = a^n / b^n
(4/2)^3 = 64/8 = 8

How to Use the Exponent Calculator

Enter any base and exponent value in the fields at the top of this page. Both fields accept positive numbers, negative numbers, and decimals. The result updates automatically as you type. For very large or very small results, the calculator automatically displays scientific notation alongside the standard form.

Understanding Exponents

An exponent (also called a power or index) indicates repeated multiplication. The expression base^exponent means the base is multiplied by itself as many times as the exponent specifies. For example, 5^4 = 5 × 5 × 5 × 5 = 625. Exponents are one of the most fundamental operations in mathematics, appearing in compound interest calculations, scientific measurement, computer science (powers of 2), and physics equations.

Negative Bases and Exponents

When the base is negative, the sign of the result depends on whether the exponent is odd or even. A negative base raised to an even exponent is always positive: (-3)^2 = 9. A negative base raised to an odd exponent is always negative: (-3)^3 = -27. Negative exponents do not make the result negative — they produce the reciprocal of the positive-exponent result. For example, 4^-2 = 1/16 = 0.0625.

Fractional Exponents and Roots

Fractional exponents represent roots. The expression a^(1/2) is the square root of a, a^(1/3) is the cube root, and so on. More generally, a^(m/n) equals the n-th root of a^m. This means you can compute any root using this calculator by entering a decimal exponent: for the cube root of 27, enter base = 27 and exponent = 0.3333 to get approximately 3.

Scientific Notation

For results larger than 999,999 or smaller than 0.0001, the calculator shows scientific notation. This format expresses numbers as a value between 1 and 10 multiplied by a power of 10. For example, 2^40 = 1,099,511,627,776 is displayed as 1.0995 × 10^12. Scientific notation is essential in fields like astronomy (distances in light-years), physics (electron charge: 1.6 × 10^-19 coulombs), and chemistry (Avogadro's number: 6.022 × 10^23).

Common Powers Table

The common powers table shows powers 2 through 10 for whatever base you enter in the main calculator. This is useful for quickly scanning how rapidly a value grows. Powers of 2 are especially important in computing: 2^8 = 256 (one byte), 2^10 = 1,024 (one kibibyte), 2^20 = 1,048,576 (one mebibyte). Powers of 10 map directly to metric prefixes: 10^3 = kilo, 10^6 = mega, 10^9 = giga.

Logarithms: The Inverse of Exponents

A logarithm answers the question: "What exponent do I need to raise the base to in order to get this number?" If 10^3 = 1,000, then log base 10 of 1,000 = 3. This calculator provides three common logarithms. Log base 10 (written log or log10) is used in the pH scale, the Richter earthquake scale, and decibel measurements. The natural logarithm (ln) uses Euler's number e (2.71828...) as the base and appears throughout calculus, probability, and continuous growth models. Log base 2 (log2) is fundamental in information theory, binary data, and algorithm complexity analysis (binary search runs in O(log2 n) time).

Using Exponents in Real Life

Compound interest uses exponents: a principal P at annual rate r for n years grows to P × (1 + r)^n. Population growth, radioactive decay, and viral spread all follow exponential models. Computer memory is measured in powers of 2. Sound intensity is measured on a logarithmic (base-10) scale. The Richter scale for earthquakes is also logarithmic, meaning a magnitude 7 earthquake releases about 32 times more energy than a magnitude 6 earthquake.

FAQ

An exponent tells you how many times to multiply a base number by itself. For example, 2^3 means 2 multiplied by itself 3 times: 2 × 2 × 2 = 8. The base is 2 and the exponent (or power) is 3.
A negative exponent means you take the reciprocal of the base raised to the positive exponent. For example, 2^-3 = 1 / (2^3) = 1/8 = 0.125. Negative exponents always produce fractions (for bases greater than 1). They do not make the result negative.
Scientific notation expresses very large or very small numbers as a coefficient multiplied by a power of 10. For example, 1,000,000 = 1 × 10^6 and 0.000001 = 1 × 10^-6. It is used in science and engineering to handle numbers too unwieldy to write out in full.
log (common logarithm) uses base 10, so log(100) = 2 because 10^2 = 100. ln (natural logarithm) uses base e (approximately 2.71828), so ln(e) = 1. Log base 10 is common in chemistry and decibels. Natural log is common in calculus, physics, and continuous growth models.
The power rule states that (a^m)^n = a^(m×n). When you raise a power to another power, you multiply the exponents. For example, (2^3)^4 = 2^12 = 4,096.