| n | 2n |
|---|---|
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
| 4 | 16 |
| 5 | 32 |
| 6 | 64 |
| 7 | 128 |
| 8 | 256 |
| 9 | 512 |
| 10 | 1,024 |
| 11 | 2,048 |
| 12 | 4,096 |
| 13 | 8,192 |
| 14 | 16,384 |
| 15 | 32,768 |
| 16 | 65,536 |
| 17 | 131,072 |
| 18 | 262,144 |
| 19 | 524,288 |
| 20 | 1,048,576 |
| n | 10n |
|---|---|
| 0 | 1 |
| 1 | 10 |
| 2 | 100 |
| 3 | 1,000 |
| 4 | 10,000 |
| 5 | 100,000 |
| 6 | 1,000,000 |
| 7 | 10,000,000 |
| 8 | 100,000,000 |
| 9 | 1,000,000,000 |
An exponent (also called a power or index) is a mathematical shorthand that tells you how many times to multiply a number by itself. In the expression bn, b is the base and n is the exponent. For example, 25 = 2 × 2 × 2 × 2 × 2 = 32. This notation is far more compact than writing out repeated multiplication, especially for large powers.
The word exponent comes from the Latin exponere, meaning "to set out" or "to explain". In the UK, you will encounter this topic at GCSE as indices (singular: index) and at A-Level as exponential functions. All three terms — exponent, power, index — refer to the same concept.
The index laws (also called laws of exponents or rules of indices) are the fundamental rules governing how powers behave. Understanding why they work, not just what they say, is key to GCSE and A-Level success.
When multiplying two powers with the same base, add the exponents. Proof: am = a × a × ... (m times), and an = a × a × ... (n times). Multiplying gives a repeated m+n times, so am+n. Example: 53 × 52 = (5×5×5) × (5×5) = 55 = 3125.
When dividing powers with the same base, subtract the exponents. Proof: am ÷ an = (a × a × ... m times) ÷ (a × a × ... n times). Cancel n copies of a, leaving m−n copies: am−n. Example: 27 ÷ 23 = 128 ÷ 8 = 16 = 24.
When raising a power to another power, multiply the exponents. Proof: (am)n means multiply am by itself n times. By the product rule, this gives am+m+...+m (n times) = amn. Example: (32)4 = 38 = 6561.
Any non-zero base raised to the power zero equals 1. This follows from the quotient rule: an ÷ an = an−n = a0, and any number divided by itself is 1. So a0 = 1. This holds for all bases except zero; 00 is undefined in general, though it is sometimes taken as 1 by convention in combinatorics.
A negative exponent means take the reciprocal. From the quotient rule: a0 ÷ an = a−n = 1/an. This does not make the result negative — it makes it a fraction. Example: 4−2 = 1/42 = 1/16 = 0.0625. For a base between 0 and 1, a negative exponent actually gives a result greater than 1: (0.5)−2 = 1/(0.5)2 = 1/0.25 = 4.
Fractional exponents connect powers to roots, which is one of the most elegant results in elementary algebra.
When working with fractional exponents on a calculator, it is often easier to calculate the root first and then raise to the power, to avoid large intermediate numbers.
Scientific notation (also called standard form) uses powers of 10 to write very large or very small numbers compactly: a × 10n, where 1 ≤ a < 10. For example, the speed of light is approximately 3 × 108 m/s, and the mass of a proton is approximately 1.67 × 10−27 kg. At GCSE, you must be able to convert between standard form and ordinary numbers, and perform arithmetic in standard form.
Powers of 10 follow the same index laws: 103 × 104 = 107. This is why the metric system is so convenient — each prefix (kilo, mega, giga) represents a power-of-10 multiplier.
Exponential growth occurs when a quantity is repeatedly multiplied by a constant factor. If a population doubles every year, after n years it is 2n times its original size. Starting with 1,000 bacteria, after 10 doublings you have 1,000 × 210 = 1,024,000.
Compound interest is the most important financial application of exponents. If you invest £P at annual interest rate r, after n years the amount is A = P × (1+r)n. Investing £1,000 at 5% for 20 years gives £1,000 × (1.05)20 = £2,653. The Rule of 72 is a useful approximation: divide 72 by the annual interest rate to estimate the doubling time. At 6%, money doubles in roughly 72/6 = 12 years.
Exponential decay is the mirror image of growth, using a base between 0 and 1. In radioactive decay, half-life T1/2 is the time for half the atoms to decay. After n half-lives, the fraction remaining is (1/2)n. Carbon-14 has a half-life of about 5,730 years, making it useful for dating organic material up to about 50,000 years old. Drug concentration in the bloodstream also decays exponentially after each dose.
The number e ≈ 2.71828 is the base of the natural exponential function. It arises naturally from compound interest: if you compound interest continuously at rate r for time t, the growth factor is ert. The function ex is its own derivative, making it the natural choice for modelling continuous growth and decay in calculus. At A-Level, you will study ex and its inverse, the natural logarithm ln(x).
The identity eiπ + 1 = 0, known as Euler's identity, is considered one of the most beautiful equations in mathematics, connecting e, i (the imaginary unit), π, 1, and 0 in a single relationship.
At GCSE, all major exam boards (AQA, Edexcel, OCR) assess indices in multiple contexts:
Higher-tier GCSE students must also handle fractional and negative indices in algebraic expressions, for example simplifying (4x2)3/2 = 8x3.
At A-Level Maths (Year 1 and Year 2), exponentials and logarithms form one of the major topics. Key content includes:
Algebraic expressions involving indices appear throughout higher mathematics. Key skills include:
An exponent (also called a power or index) tells you how many times to multiply a number by itself. In the expression 25, the base is 2 and the exponent is 5, meaning 2 × 2 × 2 × 2 × 2 = 32. Exponents are written as superscripts to the right of the base number and are used extensively in algebra, science, and computing.
Any non-zero number raised to the power of zero equals 1. So 50 = 1, 1000 = 1, and even 9990 = 1. The reasoning follows from the quotient rule: an ÷ an = an−n = a0, and any number divided by itself is 1. The special case 00 is mathematically undefined in most contexts, though sometimes taken as 1 in combinatorics.
When multiplying powers with the same base, add the exponents: am × an = am+n. For example, 32 × 33 = 35 = 243. This is the Product Rule of indices. You can only add exponents when the bases are identical — you cannot apply this rule to 23 × 53 (different bases), but you can to (2×5)3 = 103 = 1000.
A negative exponent means take the reciprocal (1 divided by). So a−n = 1/an. For example, 2−3 = 1/23 = 1/8 = 0.125. Negative exponents do not make the result negative — they make it a fraction less than 1 (for bases greater than 1). This is why very small numbers in scientific notation have negative powers of 10: 0.001 = 10−3.
A fractional exponent a1/n means the nth root of a. So 91/2 = √9 = 3, and 81/3 = ∛8 = 2. More generally, am/n = (n√a)m. For example, 82/3 = (∛8)2 = 22 = 4. This connection between fractional exponents and roots is a key topic at GCSE Higher tier and A-Level.
23 means 2 multiplied by itself 3 times: 2 × 2 × 2 = 8. The base is 2 and the exponent is 3. 32 means 3 multiplied by itself 2 times: 3 × 3 = 9. The base is 3 and the exponent is 2. The order matters greatly: swapping the base and exponent almost always gives a different answer. The only exception is when base and exponent are equal (e.g. 22 = 22).