The three most-used laws of exponents look like algebra, but they are really counting arguments about tiles. Each law is a statement about how many copies of a you end up with after stacking, nesting, or grouping. You can literally verify them by pushing little tiles around.

Below are the three tile pictures. After you have looked at them once, the three laws stop being facts to memorise and become facts you can see.

Law 1 — a^m \cdot a^n = a^{m+n}

Write a^m as a strip of m identical tiles, each labelled a. Write a^n as a strip of n identical tiles, same shape. Now glue the two strips end to end. You get a single strip of m + n tiles. That strip is a^{m+n} by definition.

Product law shown as end-to-end tile strips On the left, three tiles labelled a representing a cubed, next to a times sign, then two tiles labelled a representing a squared, then an equal sign, then five tiles shaded and labelled a representing a to the fifth. A caption below reads three plus two equals five. a a a a³ × a a a² = a a a a a a⁵ 3 + 2 = 5 copies
Three tiles next to two tiles, glued into one strip of five. The exponents on the left ($3$ and $2$) *add* to produce the exponent on the right ($5$). Nothing is happening to the value of $a$; only the *count* of $a$-tiles changes.

Why the addition is on the count: the exponent is literally a tally. When you glue a 3-tally to a 2-tally, the combined tally is 5. The operation "multiply the values" corresponds to the operation "add the tallies" because of how repeated multiplication distributes across the glue.

Law 2 — (a^m)^n = a^{m \cdot n}

Now take n copies of a^m — that is, n strips, each of length m. Stack them into a rectangle. The rectangle has n rows, each of m tiles. The total number of tiles is m \cdot n. Read it back as a single strip and you have a^{mn}.

Power of a power shown as a rectangle of tiles On the left, three rows of two tiles each, arranged in a rectangle, representing a squared raised to the third. Each tile is labelled a. On the right, six tiles in a single strip, labelled a, representing a to the sixth. A caption reads two times three equals six. a a a a a a (a²)³ — 3 rows of 2 = a a a a a a a⁶ (6 tiles) 2 × 3 = 6
A $2$-by-$3$ rectangle (three copies of $a^2$) reshaped into a single strip of $6$. The exponents *multiply*: $(a^2)^3 = a^{2 \cdot 3} = a^6$. The outer exponent tells you how many rows; the inner exponent tells you how many tiles per row; the total is the product.

Law 3 — (ab)^m = a^m \cdot b^m

The third law is about mixing two different bases. Write (ab)^m as m copies of the pair (ab), stacked. Each copy contains one a and one b. Now sort the tiles: put all the a-tiles on the left and all the b-tiles on the right. You get one strip of m a-tiles (a^m) next to one strip of m b-tiles (b^m).

Power of a product shown as sorted tiles On the left, three pairs of tiles each containing an a tile followed by a b tile, representing a b raised to the third. On the right, three a tiles grouped together followed by three b tiles grouped together, representing a cubed times b cubed. An arrow between the two configurations shows the sorting step. a b a b a b (ab)³ — ab ab ab → a a a b b b a³ · b³ (sorted) sorting $a$s left, $b$s right
$(ab)^3 = a b \cdot a b \cdot a b$. The six tiles shuffle: $a$s to the left, $b$s to the right. The number of tiles is unchanged (six), but the grouping is now $a^3 \cdot b^3$. The sorting step is what uses commutativity — it is the only place where you rely on $a b = b a$.

Why the sorting is allowed: the product a b \cdot a b \cdot a b has the two factors a and b alternating. Swapping any two adjacent letters a b \to b a does not change the product (commutativity), and swapping repeatedly lets you group all as to the left and all bs to the right. The sorted product is a \cdot a \cdot a \cdot b \cdot b \cdot b = a^3 b^3.

A concrete numerical check

Swap in real numbers (a = 2, b = 5, m = n = 3) and verify all three laws:

The tile pictures make the equalities obvious: each law is just a different way to group the same physical tiles.

Why this matters — the slogan

If you remember only one slogan from the laws of exponents, let it be this: exponents count multiplications. Every law is a statement about how different groupings of the same tiles relate to each other. Once you see the tiles, the algebra is just bookkeeping.

And once you hold the tile picture in mind, the most common traps of algebra melt:

The tile view is the fastest way to remember which laws do hold, and why the tempting-but-false ones do not.

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