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 is a single interactive tile-board. Pick a law, choose the exponents, and hit Animate to watch the tiles physically glue, stack, or sort themselves into the right-hand side of the identity. After you have seen the three animations once, the 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.
Pick Law 1 in the selector above, set any m and n, then hit Animate. The two separated strips slide together and fuse into one.
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}.
Switch to Law 2 in the selector. The left side starts as an m \times n rectangle (dashed outline); press Animate and the rectangle unspools row-by-row into a single strip of length m \cdot n.
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).
Switch to Law 3. Blue a-tiles and orange b-tiles start alternating in pairs; Animate and they sort themselves — blue tiles slide left, orange tiles slide right.
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:
- Law 1. 2^3 \cdot 2^3 = 8 \cdot 8 = 64, and 2^{3+3} = 2^6 = 64. ✓
- Law 2. (2^3)^3 = 8^3 = 512, and 2^{3 \cdot 3} = 2^9 = 512. ✓
- Law 3. (2 \cdot 5)^3 = 10^3 = 1000, and 2^3 \cdot 5^3 = 8 \cdot 125 = 1000. ✓
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:
- a^m + a^n \ne a^{m+n} — because addition of tiles is not gluing the strips; it is genuinely different.
- (a + b)^m \ne a^m + b^m — because (a + b)^m is not a rectangle of a-tiles next to a rectangle of b-tiles. It is a mixture, and the mixture has cross-terms.
- (a^m)^n is a rectangle, not a new giant tile. The outer exponent multiplies the inner one, it does not stack on top.
The tile view is the fastest way to remember which laws do hold, and why the tempting-but-false ones do not.
Related: Exponents and Powers · A Tower of 2s: Watch Doubling Explode Off the Screen by Step 30 · Why (a+b)² Is Not a² + b² — Exponents Are Not Distributive · Is 2⁻³ Really Negative Eight? The Sign-on-the-Answer Trap