An index is a tiny number that tells a bigger one to multiply itself, over and over. Do it just a few times and the result rockets off the page. Let's fold some paper and find out why.
Start hereWhen you see 2⁴, it doesn't mean 2 times 4. It means 2 × 2 × 2 × 2 — the number 2, multiplied by itself, four times over. Work it out and you get 16. That little raised 4 is doing a big job: it's counting how many 2s are in the line.
That's the entire secret of this page. A power is just a compact way to write a long chain of the same multiplication. Instead of scribbling 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2, you write 2¹⁰ and you're done. But here's the part that will surprise you: even though the writing stays small, the number it stands for does not. Repeated multiplication grows so fast that a single sheet of paper, folded a few dozen times, would reach past the edge of space. Let's build that intuition piece by piece.
Every power has two parts, and naming them makes everything else click. The base is the big number on the bottom — it's the thing being multiplied. The index (also called the exponent, or just the power) is the small number floating at the top-right — it counts how many copies of the base get multiplied together. So in 2⁴, the base is 2 and the index is 4: "use four 2s." Drag the two sliders and watch the power unpack itself into a plain multiplication.
Notice what the index really controls: it's not how big each step is, it's how many steps there are. Slide the index up by just one and you add a whole extra copy of the base to the chain — which doesn't just add a little, it multiplies the whole answer again. That's the engine behind everything below.
Three words for the same small number can trip people up, so let's settle them now: index, exponent, and power all point at that little raised figure. Your IB teacher might say "exponent," a textbook might say "index" (its plural is "indices"), and in everyday speech people just say "to the power of." They're interchangeable — don't let the vocabulary make a simple idea feel hard. And watch the two ends of the slider: when the index is 1, like 7¹, there's only one copy in the chain, so the power is just the base itself (7¹ = 7). Nothing has been multiplied yet. The growth only begins once the index climbs to 2 and beyond — which is exactly when the doubling story is about to get interesting.
Two indices are so common they earned their own nicknames. When the index is 2, we say "squared" — because 5² ("five squared") is the area of a real square that's 5 across and 5 down. When the index is 3, we say "cubed" — because 4³ ("four cubed") is the number of little blocks in a cube that's 4 wide, 4 deep, and 4 tall. For every other index we just say "to the power of": 2⁵ is "two to the power of five." Tap a power to hear it and see it unpacked.
"Squared" really means a square: 5 rows of 5 dots make a 5 × 5 square holding 25 dots in all.
There's a reason those two nicknames are about shapes. An index of 2 builds a flat, two-dimensional square, and an index of 3 builds a solid, three-dimensional cube — the words literally describe the geometry. That's also a sneaky preview of why powers grow so fast: going from squared to cubed isn't a small bump, it's adding an entire new direction to fill. A 10 × 10 square holds 100 tiles, but a 10 × 10 × 10 cube holds a thousand. One extra index, ten times more stuff. Hold on to that feeling of a number quietly multiplying out into a bigger world — because the next demo lets you fold it all the way to the sky.
Here is the demo that turns "powers grow fast" from a sentence into a jolt. Take one ordinary sheet of paper, about a tenth of a millimetre thick. Fold it in half and you've got 2 layers. Fold again: 4. Again: 8, then 16, 32… Each fold doubles the thickness, so after n folds the stack is 2ⁿ layers tall. The doubling starts off looking gentle and harmless. Keep dragging — and watch it leave the building, the mountains, and the planet behind.
The orange column climbs a squashed ruler where each step up means ten times taller — that's the only way to fit a fingernail and the edge of space on one screen. Even on that merciful, squashed scale the climb is dizzying. In real life you can't actually fold paper this many times, but the maths is honest: at 30 folds, one sheet's worth of doubling stands taller than the sky. That's doubling thirty times, written simply as 2³⁰.
Sit with how strange this is. You never multiplied by a big number. You only ever doubled — the smallest interesting jump there is. Yet because each fold builds on the last total instead of the original sheet, the growth feeds on itself and runs away. This is the headline fact about powers: repeated multiplication doesn't add up, it piles up. The next demo puts that runaway growth in a head-to-head race so you can feel exactly how fast it pulls ahead.
It's easy to mix up adding the same number again and again with multiplying by the same number again and again. They sound similar, but they live in different worlds. Let's race them. Both runners start at 2. One adds 2 every step (plain, steady growth — that's multiplication's slower cousin). The other doubles every step (that's a power building up). They tie for a moment at the start… then the doubler vanishes over the horizon.
For the first step or two the lines sit almost on top of each other — which is exactly why doubling feels safe at first and fools people. But adding only ever walks; doubling sprints, then flies. The brown line is what "+ 2 each time" looks like (a straight ramp). The orange line is what a power looks like (a curve that turns almost straight up). Same start, wildly different destiny.
Adding stacks one brick at a time. A power copies the whole wall, then copies that, then copies that. Small steps, taken on top of each other, become a giant leap.
Here's the cleanest way to feel the difference. With adding, every step is the same size — you tack on 2, then another 2, then another 2, forever, so the line climbs at a steady, boring angle. With doubling, every step is as big as everything you've built so far: your next jump is the height of your whole current pile. So the jumps get bigger because the pile gets bigger, and the pile gets bigger because the jumps got bigger. That loop is what bends the orange line skyward. People call this exponential growth — "exponential" being just the grown-up word for "driven by an exponent" — and it's the reason a quiet doubling can blindside you. It looks lazy for ages, then erupts all at once.
Once you see a power as "a line of identical numbers being multiplied," some neat shortcuts almost fall out by themselves. These are the laws of indices, and you never have to memorise them as magic — you can rebuild each one in your head by counting copies. Pick a base, pick two indices, and choose a rule. The coloured blocks below show what's really happening.
Count the blocks and the rules stop being mysterious. Multiply two powers of the same base? Line up all the copies and add the indices. Divide? Cancel matching copies top and bottom, so you subtract. Raise a power to a power? You're making copies of copies, so you multiply. One warning: these only work when the base is the same on both sides.
There's a lovely bonus hiding in the divide rule. Set both indices equal — say 2³ ÷ 2³. Every copy on top cancels a copy on the bottom and nothing is left… yet the answer to "a number divided by itself" is plainly 1. The index arithmetic agrees: 3 − 3 = 0, so 2³ ÷ 2³ = 2⁰. Both stories give the same result, which is why mathematicians define any base to the power of 0 as 1. It isn't a random rule someone invented — it's the only value that keeps the pattern honest.
Our whole number system is secretly built on one base: ten. And powers of ten are the friendliest powers of all, because the index simply counts the zeros. 10² is 100 (two zeros). 10³ is 1,000 (three zeros). 10⁶ is a million. Slide along and watch the zeros pile up — and meet the everyday names we give them.
This is the trick behind standard form (sometimes called scientific notation) — the tidy way scientists write gigantic and tiny numbers without an exhausting parade of zeros. Instead of writing 3,000,000 you write 3 × 10⁶; the power of ten carries all the size, and the "3" tells you the rest. A number like the distance to the Sun fits in a few characters this way. We won't go deep here, but now you can see where it comes from: it's powers of ten doing the heavy lifting.
This is the single most common slip with powers, and almost everyone makes it once. The little 4 sits up high like a friendly hint, and your brain wants to read it as "2 times 4," which would be 8. But that's not what the notation means. The raised number is a counter, not a multiplier. It says "write down four 2s and multiply them all together." Look at the two readings side by side:
Reading the index as "multiply the base by the index." Tempting, tidy — and wrong.
The index counts how many 2s to multiply. Four 2s, all multiplied, give 16.
A quick way to never fall for it again: whisper the words. "Two to the power of four" should make you say "two, two, two, two — multiply." If you ever catch yourself doing "base times index," stop and unpack the chain instead. The gap only gets wider as the index grows: 2⁴ is 16, not 8; 2¹⁰ is 1,024, not 20. By the time you reach the paper-folding numbers, the difference between the two readings is the difference between a coin and a skyscraper.
Powers aren't a classroom curiosity. Anything that repeatedly multiplies — doubling on the way up, or halving on the way down — is a power in disguise. Once you know the pattern, you start spotting it everywhere.
An old story tells of an inventor who asked a king for one grain of rice on the first square of a chessboard, two on the next, four on the next, doubling all the way to the 64th square. It sounds humble. But that's 2 doubling on top of itself square after square — a power — and long before the last square the king couldn't possibly pay. The exact total isn't the point; the point is that doubling sixty-odd times turns "one grain" into a mountain that buries the whole kingdom. That's the same engine you felt folding paper.
Many bacteria grow by splitting in two: 1 becomes 2, 2 becomes 4, 4 becomes 8. After n rounds you have about 2ⁿ of them — which is why a tiny invisible speck can become a thriving colony surprisingly fast.
Every fold doubles the layers, exactly as you saw above. The same doubling shows up when something keeps copying itself — a rumour shared with two friends who each tell two more, spreading by powers of 2.
Powers can shrink, too. Cut a quantity in half again and again — ½, ¼, ⅛, 1⁄16 — and you're using powers running downhill. Echoes fading, a hot drink cooling, light dimming through water: each step is the last one halved.
Savings that earn interest grow by multiplying a little each year, not adding the same amount. Over many years that small repeated multiplication — a gentle power — quietly builds far more than plain adding ever would.
Computer memory is built from switches that are on or off, so its sizes come in powers of 2: 2, 4, 8, 16, 32, 64, 128, 256… Those familiar "round" tech numbers are really just 2 raised to a power.
2⁴ means 2 × 2 × 2 × 2. The base is what's multiplied; the index counts how many copies. Never "2 × 4."
Each step builds on the last total, so the growth feeds itself. One sheet, folded, climbs past the sky.
Multiply → add the indices. Divide → subtract. Power of a power → multiply. And anything to the power 0 is 1.