The Laws of Indices

Powers pile up fast in the real world — computer memory doubles from kilobytes to megabytes to gigabytes, a virus multiplies each hour, interest compounds year on year. To handle numbers that big without drowning in zeros, engineers and scientists lean on a few slick rules for combining powers.

Suppose someone asks you for 2^{20} \times 2^{15}. Writing that out the long way means scribbling twenty twos, then fifteen more, and multiplying the whole lot together — a wall of numbers, and a headache waiting to happen.

Here is the good news: you never have to. A tiny handful of rules — the laws of indices — turn scary-looking towers of powers into one-line arithmetic. That giant product is simply 2^{20+15} = 2^{35}. You added the indices and you were done. This page is about why that shortcut works and how to use it without slipping up.

Everything here needs the powers to share the same base — the number being raised, like the 2 in index notation. Keep that in mind; it's the one thing everybody forgets.

Law 1 — multiplying: add the indices

When you multiply two powers of the same base, you add the indices:

a^m \times a^n = a^{m+n}

Why? A power is just a shorthand for repeated multiplication, so this is nothing more than counting the factors. Take 2^3 \times 2^4:

\underbrace{(2 \times 2 \times 2)}_{3\text{ twos}} \times \underbrace{(2 \times 2 \times 2 \times 2)}_{4\text{ twos}} = \underbrace{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2}_{7\text{ twos}} = 2^{7}

Three twos next to four twos is seven twos in a row — so 2^3 \times 2^4 = 2^{3+4} = 2^{7}. Let's check it against the real numbers so you trust the rule: 2^3 = 8 and 2^4 = 16, and 8 \times 16 = 128. Now 2^7 = 128 too. The shortcut and the long way agree — every time.

Law 2 — dividing: subtract the indices

When you divide two powers of the same base, you subtract the indices:

a^m \div a^n = a^{m-n}

Dividing is multiplying's undo button, so it's no surprise the rule flips from add to subtract. Picture the factors as a fraction, and watch the copies on the bottom cancel copies on the top. For 5^6 \div 5^2:

\frac{5 \times 5 \times 5 \times 5 \times 5 \times 5}{5 \times 5} = 5 \times 5 \times 5 \times 5 = 5^{4}

Two fives on the bottom rub out two of the six on top, leaving four — so 5^6 \div 5^2 = 5^{6-2} = 5^{4}. A quick sanity check with smaller numbers: 2^5 \div 2^3 should be 2^{2} = 4, and indeed 32 \div 8 = 4.

Law 3 — a power of a power: multiply the indices

What if a power is itself raised to a power, like (3^2)^3? Raising a^m to the power n means writing n copies of a^m and multiplying them. By Law 1 those indices all add up — n lots of m — which is the same as multiplying:

(a^m)^n = a^{mn}

So (3^2)^3 = 3^2 \times 3^2 \times 3^2 = 3^{2+2+2} = 3^{2 \times 3} = 3^{6}. Check it: 3^2 = 9, and 9^3 = 729; meanwhile 3^6 = 729.

For powers of the same base a: These only work when the base is the same: there is no such shortcut for 2^3 \times 3^2.

Combining the laws in one go

Real problems mix the laws together. The trick is to take them one step at a time, top and bottom. Simplify:

\frac{7^4 \times 7^5}{7^3}

First deal with the top using Law 1 — 7^4 \times 7^5 = 7^{4+5} = 7^{9} — so the whole thing becomes 7^9 \div 7^3. Now Law 2 subtracts: 7^{9-3} = 7^{6}. One expression, two laws, no giant products.

Here's a tougher blend that uses all three. Simplify (x^2)^3 \times x^4. Start inside the bracket with Law 3: (x^2)^3 = x^{2 \times 3} = x^{6}. Then Law 1 finishes it: x^6 \times x^4 = x^{6+4} = x^{10}. Notice we multiplied for the bracket and added for the product — using the right law at the right moment is the whole game.

Where you'll actually meet this

These laws aren't just exam gymnastics — they run through the machine you're reading this on. Computer memory is measured in powers of two: 2^{10} bytes make a kilobyte, 2^{20} bytes a megabyte, 2^{30} a gigabyte. How many kilobytes in a gigabyte? You don't multiply out those monstrous numbers — you just subtract indices:

\frac{2^{30}}{2^{10}} = 2^{30-10} = 2^{20} = 1{,}048{,}576

A little over a million — and you got there with a single subtraction. The same shortcut appears anywhere quantities double or grow in steps: a colony of bacteria splitting every hour, the squares on a chessboard doubling grain by grain, the notes in a musical octave. Whenever you're multiplying repeated copies of the same thing, the laws of indices are the fast lane.

Two traps snare almost everyone:

"Add the indices when you multiply" is a small rule with an enormous history. Because standard form writes every number as something times a power of ten, multiplying two huge (or two tiny) numbers just means adding the powers of ten — the laws of indices are exactly why that works so smoothly.

This same idea — that multiplying can be swapped for adding — is the whole reason logarithms were invented. Before calculators, astronomers and engineers had to multiply monstrous numbers by hand, and mistakes were deadly (imagine getting a ship's position wrong at sea). Logarithms turn every multiplication into a simple addition. For three centuries, from Kepler's planetary tables to the slide rules that flew on the Apollo missions, people did their hardest sums by adding indices off printed tables. A rule you learn in an afternoon once ran the world.