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:
This calculator uses the current UK grading system and educational standards to help students, parents, and teachers understand academic performance. UK education follows specific grading frameworks that differ between GCSEs, A-Levels, and university degrees.
Understanding how grades are calculated and what they mean for future progression is important for making informed decisions about subject choices, university applications, and career planning.
GCSEs in England use the 9-1 grading scale, where 9 is the highest and 4 is a standard pass (equivalent to the old C grade). A-Levels use the A*-E scale with UCAS tariff points ranging from 56 (A*) to 16 (E). Universities typically require grades between AAA and CCC depending on the course and institution, with highly competitive courses often asking for A*A*A.
A student achieving grades of A*, A, and B at A-Level would earn UCAS tariff points of 56 + 48 + 40 = 144 points total. This exceeds the typical entry requirement for most Russell Group universities (128 points or AAB equivalent) and would make the student competitive for many courses.
Source: Based on Ofqual and UCAS 2025/26 guidelines. Last updated March 2026.
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.
Data verified against official UK government sources. Last checked April 2026.