Enter the base and exponent (index) to calculate the result with full working:
In mathematics, an index (plural: indices) — also called a power or exponent — is a shorthand notation that tells you how many times to multiply a number (the base) by itself. The expression an means "a multiplied by itself n times", where a is the base and n is the index.
Example: 23 = 2 × 2 × 2 = 8. Here, 2 is the base and 3 is the index.
Indices are fundamental to algebra, calculus, scientific notation, computing (binary and storage sizes), and financial mathematics (compound interest). Mastering the laws of indices is a core requirement for GCSE Mathematics and is extended significantly at A-Level.
Example 1 — Law 1 (Multiplication):
Simplify 52 × 56. Both terms have base 5, so add the indices: 52+6 = 58 = 390,625.
Example 2 — Law 2 (Division):
Simplify 47 ÷ 43. Both terms have base 4, so subtract the indices: 47−3 = 44 = 256.
Example 3 — Law 5 (Negative Index):
Evaluate 3−2. Apply the negative index law: 3−2 = 1/32 = 1/9 ≈ 0.1111. Remember: a negative index does NOT produce a negative result — it produces a reciprocal.
Example 4 — Law 7 (Fractional Index):
Evaluate 272/3. Apply law 7: find the cube root of 27 first: 3√27 = 3. Then raise to the power 2: 32 = 9. So 272/3 = 9.
Powers of 2 are the foundation of computer science and digital storage. Every bit in a computer represents a power of 2:
| Expression | Value | Computing Context |
|---|---|---|
| 20 | 1 | 1 bit = 0 or 1 |
| 21 | 2 | 2 values |
| 28 | 256 | 1 byte (8 bits) |
| 210 | 1,024 | 1 kilobyte (approx.) |
| 216 | 65,536 | 16-bit colour depth |
| 220 | 1,048,576 | 1 megabyte (exact) |
| 230 | 1,073,741,824 | 1 gigabyte (exact) |
| 232 | 4,294,967,296 | 32-bit integer max |
| 264 | ~1.84 × 1019 | 64-bit integer max |
Scientific notation uses powers of 10 to express very large or very small numbers compactly. A number in scientific notation has the form a × 10n, where 1 ≤ a < 10 and n is an integer.
To multiply in scientific notation, multiply the decimal parts and add the powers of 10: (3 × 104) × (2 × 103) = 6 × 107.
Negative exponents represent reciprocals, not negative values. This is a critical distinction:
Fractional exponents represent roots. The denominator of the fraction tells you which root to take:
An index (plural: indices) is the power to which a base number is raised. In 23, the base is 2 and the index is 3. The expression means 2 × 2 × 2 = 8. Indices are also called exponents or powers. The terminology varies: UK school maths typically uses "indices", while international mathematics more commonly uses "exponents".
A negative index means the reciprocal of the positive power: a−n = 1/an. Example: 2−3 = 1/23 = 1/8 = 0.125. A negative index does not make the result negative — it means "one divided by the positive power". This follows from the division law: 20/23 = 20−3 = 2−3 = 1/8.
A fractional index represents a root. a1/n = the nth root of a. So 271/3 = 3√27 = 3. For a general fractional index m/n: am/n = (n√a)m. Example: 163/4 = (4√16)3 = 23 = 8. It is usually easier to take the root first to keep numbers smaller before applying the power.
Any non-zero number raised to the power of 0 equals 1: a0 = 1. This follows from the division law: an/an = an−n = a0, and any number divided by itself equals 1. So a0 = 1. Examples: 50 = 1, 1000 = 1, (−7)0 = 1. The case of 00 is mathematically indeterminate.
Add the exponents: am × an = am+n. Example: 32 × 35 = 37 = 2,187. This only works when the bases are identical. You cannot simplify 23 × 52 this way — the bases (2 and 5) are different. Worked check: 32 = 9, 35 = 243, 9 × 243 = 2,187 = 37.
Scientific notation expresses numbers as a × 10n where 1 ≤ a < 10. It is used for very large numbers (the distance to a star: 4.24 × 1016 metres) and very small numbers (an electron's mass: 9.11 × 10−31 kg). To convert: move the decimal point until you have a number between 1 and 10, counting how many places you moved (positive for large, negative for small).
Because a negative number multiplied by a negative number gives a positive: (−2)2 = (−2) × (−2) = +4. With an even exponent, the negatives pair up and cancel. With an odd exponent, one negative is always left unpaired: (−2)3 = (−2) × (−2) × (−2) = 4 × (−2) = −8. The rule: negative base with even exponent = positive; negative base with odd exponent = negative.