Five simple rules turn towering repeated multiplication into one line of arithmetic, and explain why a non-zero number to the power of zero is always 1.
An exponent is shorthand for repeated multiplication. But once two exponent expressions have to be combined, five specific laws take over. Each law has its own exact rule for what to do with the exponents. This article explains where each law comes from, so you can remember it, and shows how exponents describe money, bacteria, computer memory and the very large and very small numbers in science.
An exponent (also called a power or an index) tells you how many times to multiply a number by itself. In 24, the base is 2 and the exponent is 4, meaning 2 × 2 × 2 × 2.
24 = 2 × 2 × 2 × 2 = 16. Read aloud as “two to the power of four” or “two to the fourth”. Notice that 24 is not 2 × 4 = 8. The exponent counts how many 2s are multiplied, it is not itself multiplied.
This is the same idea already covered for squares (n²) and cubes (n³) in Squares, Square Roots, Cubes & Cube Roots. Exponents simply extend that pattern to any power, not just 2 or 3. The table below shows how quickly powers grow.
| Power | Powers of 2 | Powers of 3 | Powers of 10 |
|---|---|---|---|
| Exponent 1 | 2 | 3 | 10 |
| Exponent 2 | 4 | 9 | 100 |
| Exponent 3 | 8 | 27 | 1,000 |
| Exponent 4 | 16 | 81 | 10,000 |
| Exponent 5 | 32 | 243 | 100,000 |
| Exponent 6 | 64 | 729 | 1,000,000 |
Any non-zero number raised to the power of 0 equals exactly 1.
50 = 1. 1000 = 1. Even 999,9990 = 1.
Why is this true? Look at the pattern in the powers of 2 as the exponent goes down by 1 each time. Each step divides by 2:
23 = 8, 22 = 4, 21 = 2, 20 = 1, 2-1 = 1/2, 2-2 = 1/4 …
The pattern only continues smoothly if 20 = 1. Another way to see it: any number divided by itself is 1, so 23 ÷ 23 = 1. By the quotient law (below) this is 23−3 = 20. So 20 must be 1.
One special case: 00 is not defined in school mathematics, so the rule is for non-zero bases only.
A negative exponent does not make the result negative. It means “take the reciprocal”, flipping the number into a fraction. In general, a-n = 1 ÷ an.
(a) 2-3 = 1/23 = 1/8 = 0.125. The result is a small positive fraction, not a negative number.
(b) 10-3 = 1/1,000 = 0.001. This is how scientific notation writes small numbers (see Scientific Notation).
(c) (2/3)-2: flip the fraction and make the power positive: (3/2)2 = 9/4 = 2.25.
This is one of the most common exponent mistakes: assuming the negative sign carries into the final answer.
The fix: a negative exponent means “flip to a fraction”, not “make the answer negative”. 2-3 = 1/23 = 1/8, a positive value less than 1.
When multiplying two powers with the same base, add the exponents.
23 × 24 = 23+4 = 27 = 128. Check directly: 8 × 16 = 128. ✓
Why it works: 23 means three 2s multiplied together, and 24 means four 2s. Multiply them and you have 3 + 4 = 7 twos in a row.
When dividing two powers with the same base, subtract the exponents.
25 ÷ 22 = 25−2 = 23 = 8. Check directly: 32 ÷ 4 = 8. ✓
Why it works: 25 ÷ 22 = (2×2×2×2×2) ÷ (2×2). Two of the 2s cancel from top and bottom, leaving three 2s.
When a power is itself raised to another power, multiply the exponents together.
(23)2 = 23×2 = 26 = 64. Check directly: 82 = 64. ✓
Why it works: (23)2 means 23 × 23, which is two groups of three 2s, so six 2s in total.
This mixes up Law 1 (add, for multiplying same-base powers) with Law 3 (multiply, for a power raised to another power). They are two different situations with two different rules.
The fix: 32 × 33 = 32+3 = 35 = 243. The exponents add because this is two separate powers being multiplied together, not one power raised to another. Multiplying exponents only applies to expressions like (32)3, where a power itself gets raised to a further power.
When a product of two numbers is raised to a power, the exponent applies to each factor separately.
(2×3)3 = 23 × 33 = 8 × 27 = 216. Check directly: 63 = 216. ✓
With letters: (2x3)2 = 22 × (x3)2 = 4x6. The number 2 is squared too, so the answer is 4x6, not 2x6.
The same idea applies to division: the exponent applies separately to the numerator and the denominator.
(6÷2)3 = 63 ÷ 23 = 216 ÷ 8 = 27. Check directly: 33 = 27. ✓
This is a very common overreach: applying the power-of-a-product law to addition, where it simply does not hold.
The fix: the “distribute the exponent” laws only work for multiplication and division, never for addition or subtraction. (3+4)2 = 72 = 49, but 32+42 = 9+16 = 25. They are completely different answers. There is no exponent law for the power of a sum. Expanding (a+b)2 properly means multiplying it out as (a+b)×(a+b), a topic covered in algebra.
| Law | Rule |
|---|---|
| Product of Powers | am × an = am+n |
| Quotient of Powers | am ÷ an = am−n |
| Power of a Power | (am)n = am×n |
| Power of a Product | (ab)n = anbn |
| Power of a Quotient | (a÷b)n = an÷bn |
Plus two special results: a0 = 1 (for a ≠ 0) and a-n = 1 ÷ an. Remember that Laws 1 and 2 need the same base. You cannot combine 23 × 34 into a single power, because the bases are different.
Tap any expression below to see which law applies and how it simplifies, step by step.
Tap an expression above to see the full breakdown.
Harder questions use several laws together. Work from the inside out, and change numbers to the same base when you can.
(a) (23 × 22) ÷ 24: the top is 23+2 = 25. Then 25 ÷ 24 = 25−4 = 21 = 2.
(b) (32)3 × 3-4: first (32)3 = 36. Then 36 × 3-4 = 36−4 = 32 = 9.
(c) (25 × 4) ÷ 23: write 4 as 22. Then 25 × 22 = 27, and 27 ÷ 23 = 24 = 16.
(d) 12a5 ÷ 3a2: divide the numbers (12 ÷ 3 = 4) and subtract the powers (5 − 2 = 3). The answer is 4a3.
The brackets matter. Without brackets, the exponent applies only to the 3, not to the minus sign.
The fix: −32 means −(32) = −9. But (−3)2 = (−3) × (−3) = +9. A negative number squared is positive only when the brackets show that the minus sign is part of the base. In the same way, −24 = −16 but (−2)4 = 16.
Powers of 10 are the exponent laws in their simplest form. They let you multiply large and small numbers quickly: (3 × 104) × (2 × 103) = 6 × 104+3 = 6 × 107. This is exactly how scientific notation works, covered in Scientific Notation: Writing Very Large and Very Small Numbers.
Roots can also be written as exponents. A power of 1/2 means a square root, and a power of 1/3 means a cube root. So 91/2 = √9 = 3 and 81/3 = ∛8 = 2. A power like 163/4 means “take the fourth root, then cube it”: the fourth root of 16 is 2, and 23 = 8. These fit with all the laws above, and you will meet them in more detail in later algebra.
Computer memory is traditionally measured in powers of 2, as in Number Base Systems: 210 = 1,024 bytes make a kilobyte, and 220 = 1,048,576 bytes make a megabyte. (Some manufacturers use 1,000 bytes for a kilobyte instead, and the name kibibyte was created for 1,024 bytes, so always check which meaning is being used.) Bacteria that double every hour under ideal conditions also grow by powers: starting from 1, after 10 hours there are 210 = 1,024 bacteria.
Rs 10,000 is saved for 3 years at 8% a year, with the interest added each year. Each year the money is multiplied by 1.08, so after 3 years it is 10,000 × 1.083 = 10,000 × 1.259712 = Rs 12,597.12. The exponent counts the years. (These are practice numbers, not real bank rates.)
A 64 g sample loses half of its mass every day. After 5 days, how much is left?
Remaining = 64 × (1/2)5 = 64 × 2-5 = 64 ÷ 32 = 2 g. Check by halving: 64, 32, 16, 8, 4, 2. ✓
A message is sent to 2 people. Each of them forwards it to 2 new people, and so on. How many people receive it in round 10, and how many have received it altogether?
Round 10: 210 = 1,024 people.
Altogether: 2 + 4 + 8 + … + 1,024 = 2,046 people. This equals 211 − 2 = 2,048 − 2. Within only ten rounds, a message can reach thousands of people.
Imagine putting one grain of rice on the first square of a chessboard, two on the second, four on the third, and so on, doubling each time. On the 64th square alone you would need 263 grains, which is about 9.2 quintillion. The total across all 64 squares is 264 − 1, about 1.8 × 1019 grains. This is the classic “wheat and chessboard problem”, a famous way to show how explosively powers grow.
Tap the correct answer for each question. Your score is tracked as you go, and your best score is saved on this device.
1. 5-2 = 1/52 = 1/25 = 0.04.
2. 24+(−2) = 22 = 4.
3. (−2)3 = (−2)(−2)(−2) = −8. And −24 = −(24) = −16.
1. 2-4 = 1/24 = 1/16 = 0.0625.
2. Top: 35+2 = 37. Then 37−4 = 33 = 27.
3. 24 × (x3)4 = 16x12.
4. Write 8 = 23 and 4 = 22. Then 23 × 25 = 28, and 28 ÷ 22 = 26 = 64.
5. Amount = 20,000 × 1.053 = 20,000 × 1.157625 = Rs 23,152.50.
Students remember the laws better when they discover why they work. Ask learners to write 23 × 24 in full, count the 2s, and say what they notice. Do the same for division by crossing out matching factors. Encourage them to check every answer with small numbers, as the examples above do. Common errors are multiplying the base, treating a negative exponent as a negative number, and distributing a power over addition. Real contexts such as savings growth, bacteria and computer memory make the idea more concrete.
Exponent laws are used constantly in scientific notation, compound growth calculations, computer science and, very soon in your maths journey, algebra, where these exact same five laws apply to expressions with letters instead of plain numbers. Getting comfortable with them now, using numbers you can double-check by hand, makes every later use of exponents dramatically easier.
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