Calculate powers, roots, and scientific notation. Supports negative exponents, fractional exponents, and exponential growth curves. Live results for USA and UK.
Exponents are a fundamental mathematical operation expressing repeated multiplication. When we write 2^10 (2 to the power of 10), we mean 2 multiplied by itself 10 times: 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1,024. Exponents appear throughout mathematics, science, engineering, finance, and computing — from compound interest calculations to the binary number system, from atomic physics to the Richter scale for earthquakes.
In the USA, powers are written as b^n or bn. In the UK (and internationally), the same notation is used in mathematics, but in British secondary education the term "index" (plural "indices") is more commonly used than "exponent." The rules are identical regardless of terminology.
| Rule | Formula | Example |
|---|---|---|
| Product Rule | b^m × b^n = b^(m+n) | 2^3 × 2^4 = 2^7 = 128 |
| Quotient Rule | b^m ÷ b^n = b^(m−n) | 2^6 ÷ 2^2 = 2^4 = 16 |
| Power of a Power | (b^m)^n = b^(m×n) | (2^3)^2 = 2^6 = 64 |
| Zero Exponent | b^0 = 1 | 5^0 = 1 |
| Negative Exponent | b^−n = 1/b^n | 2^−3 = 1/8 = 0.125 |
| Fractional Exponent | b^(1/n) = n√b | 8^(1/3) = ∛8 = 2 |
| Fractional Exponent | b^(m/n) = (n√b)^m | 8^(2/3) = 4 |
A negative exponent means taking the reciprocal: b^−n = 1/b^n. For example, 10^−3 = 1/1000 = 0.001. This is used extensively in scientific notation for very small numbers. The mass of a proton is approximately 1.67 × 10^−27 kg — the −27 exponent means the number is 27 decimal places to the right of the decimal point.
A fractional exponent represents a root. b^(1/2) = √b, b^(1/3) = ∛b, and generally b^(1/n) = n√b. Combining with an integer: b^(m/n) = (n√b)^m. For example, 27^(2/3) = (∛27)^2 = 3^2 = 9. This calculator handles all fractional exponent inputs.
Scientific notation expresses very large or very small numbers as a coefficient (between 1 and 10) multiplied by a power of 10. In the USA it is called scientific notation; in the UK GCSE and A-Level curriculum it is called standard form.
Format: a × 10^n, where 1 ≤ a < 10
Examples: The speed of light is 3 × 10^8 m/s. The diameter of a human hair is approximately 7 × 10^−5 m. Avogadro's number is 6.022 × 10^23.
| Number | Scientific Notation | UK Standard Form |
|---|---|---|
| 1,000,000 | 1 × 10^6 | 1 × 10^6 |
| 0.00045 | 4.5 × 10^−4 | 4.5 × 10^−4 |
| 6,000,000,000 | 6 × 10^9 | 6 × 10^9 |
| 0.0000001 | 1 × 10^−7 | 1 × 10^−7 |
The compound interest formula A = P(1 + r/n)^(nt) uses an exponent to model exponential growth of investments. If $10,000 is invested at 7% annual interest compounded annually for 20 years, the calculation is: A = 10,000 × (1.07)^20 = 10,000 × 3.8697 = $38,697. The power of compounding over time is literally the power of exponents — small differences in the exponent (time or rate) create large differences in final results.
In computing, powers of 2 are fundamental because computers store data in binary (base 2). One bit holds 2^1 = 2 values. One byte holds 2^8 = 256 values. A 32-bit system can address 2^32 = 4,294,967,296 memory locations. A 64-bit system can address 2^64 ≈ 1.8 × 10^19 locations.
| Power | Value | Common Name |
|---|---|---|
| 2^10 | 1,024 | 1 Kilobyte (approx) |
| 2^20 | 1,048,576 | 1 Megabyte (approx) |
| 2^30 | 1,073,741,824 | 1 Gigabyte (approx) |
| 2^40 | 1,099,511,627,776 | 1 Terabyte (approx) |
A negative exponent means take the reciprocal: b^−n = 1/b^n. For example, 2^−3 = 1/8 = 0.125. Negative exponents are used in scientific notation for very small numbers like 10^−9 (nanometres).
A fractional exponent represents a root. b^(1/2) = √b, b^(1/3) = ∛b. For example, 64^(1/2) = 8 and 27^(1/3) = 3. The general rule is b^(m/n) = (n-th root of b)^m.
Scientific notation writes numbers as a coefficient (1 to 9.999...) multiplied by a power of 10. In the UK curriculum it is called standard form. Example: 45,000 = 4.5 × 10^4.
Any non-zero number raised to the power of 0 equals 1. This is because b^n ÷ b^n = b^(n−n) = b^0, and any number divided by itself equals 1.
Compound interest uses the formula A = P(1+r)^n where n is time in years. The exponent represents how many compounding periods occur, making it possible to model exponential growth of investments.
They mean the same thing. In the UK GCSE and A-Level curriculum, "index" (plural: indices) is the standard term. In US math education, "exponent" is the standard term. The notation b^n and all rules are identical.
10^n simply places n zeros after the 1 (for positive n). 10^6 = 1,000,000. For negative n, it places the decimal point n places to the left: 10^−4 = 0.0001. This makes powers of 10 useful for scientific notation.
Exponential growth occurs when a quantity increases by the same percentage each period. If a population grows at 3% per year, after n years it is P × (1.03)^n. Exponential growth starts slowly but accelerates dramatically — this is why compound interest and viral spread both follow this pattern.
Results are for educational purposes. Very large or very small numbers may be subject to floating-point precision limits. For financial calculations always consult a qualified professional.