Key facts
- 210 = 1,024; 53 = 125; 106 = 1,000,000.
- Any non-zero number to the power 0 is 1.
- Negative exponents mean reciprocal: 2−3 = 1/8 = 0.125.
- Fractional exponents are roots: 81/3 = 2.
An exponent says how many times to multiply a number by itself — and it reaches far beyond whole numbers. This free exponent calculator raises any base to any power: positive, negative, zero or fractional.
Enter the base and the exponent to get the result instantly, with the exponent laws summarised below for homework and revision.
Quick answer: An exponent tells you how many times to multiply the base by itself — 2^10 = 1024. Negative exponents give fractions (2^−3 = 1/8) and fractional exponents give roots (9^0.5 = 3). Enter the base and exponent below.
How to use
- Enter the base (the number being multiplied).
- Enter the exponent (how many times — can be negative or fractional).
- The result appears instantly.
- Try the laws in the table below to build intuition.
Exponent laws
| Term | Meaning |
|---|---|
| am × an = am+n | Same base: add the exponents. 23 × 24 = 27 = 128. |
| (am)n = amn | Power of a power: multiply exponents. (23)2 = 26 = 64. |
| a−n = 1/an | Negative exponent = reciprocal. 2−3 = 1/8. |
| a0 = 1 | Anything (except 0) to the power zero is 1. |
| a1/n = ⁿ√a | Fractional exponents are roots: 81/3 = 2. |
These laws are why (23)2 = 26 but 232 = 29 = 512 — stacked exponents evaluate top-down.
Worked examples
Whole-number powers
210: 2 multiplied by itself 10 times = 1,024 — the number behind kilobytes and computing.
53 = 5 × 5 × 5 = 125; 106 = 1,000,000.
Tricky exponents
2−3 = 1 ÷ 23 = 1/8 = 0.125.
91/2 = √9 = 3 — a half exponent is a square root. 70 = 1.
Powers of 2 worth knowing
| Power | Value | Shows up in |
|---|---|---|
| 28 | 256 | Values in one byte |
| 210 | 1,024 | Kilobyte |
| 216 | 65,536 | Old colour depths |
| 220 | 1,048,576 | Megabyte |
| 230 | 1,073,741,824 | Gigabyte |
| 232 | 4,294,967,296 | 32-bit limit |
Exponents in real life
Compound interest runs on exponents: $1,000 invested at 5% annual interest for 10 years grows to 1,000 × 1.0510 = $1,628.89. The same mathematics describes population growth, radioactive decay and the Richter scale for earthquakes.
Frequently asked questions
What is 0 to the power 0?
Indeterminate in general mathematics — the calculator treats 00 as undefined. (Some contexts define it as 1 for convenience.)
How do fractional exponents work?
a1/n is the nth root of a: 81/3 = 2, 91/2 = 3. Combine: am/n = (ⁿ√a)m.
Can the base be negative?
Yes for whole-number exponents: (−2)3 = −8. Fractional powers of negatives leave the real numbers — the calculator flags those.
Why does anything to the power 0 equal 1?
From the laws: am ÷ am = a0, but anything divided by itself is 1.
What is the difference between 2^3^2 and (2^3)^2?
Stacked exponents evaluate top-down: 2^(3^2) = 29 = 512, while (23)2 = 82 = 64. Brackets change everything.
How do you calculate exponents step by step?
An exponent tells you how many times to multiply the base by itself. For 3 to the power of 4, multiply 3 x 3 x 3 x 3 to get 81. A negative exponent means the reciprocal: 2 to the power of -3 equals 1 divided by 8. Any nonzero number to the power of 0 equals 1.
How do you calculate the power of a number?
Raise a number to a power by multiplying the base by itself as many times as the exponent says: 5 to the power of 3 equals 5 x 5 x 5, or 125. On a calculator use the x^y or caret key; in Excel type =POWER(5,3).
How do I find the exponent if I only know the answer?
Use logarithms: ax = y rearranges to x = log(y) ÷ log(a), with any consistent base. For example, 2x = 1,024 gives x = log(1,024) ÷ log(2) = 10. Try it on the Logarithm Calculator.
Sources
- Khan Academy — exponent properties lessons
Key takeaways
- am×an = am+n; (am)n = amn; a−n = 1/an.
- Anything0 = 1; fractional exponents are roots.
- Stacked powers evaluate top-down: 2^3^2 = 512, not 64.
Comments
Leave a comment