We already know that a power is shorthand for repeated multiplication: x^3 means x \times x \times x. With letters the same rules apply, and they let us combine powers of the same base without writing everything out.

The big idea is to count the factors. Multiply two powers of x and just count how many x's you have:

Two factors and three factors make five factors in total — so we add the indices. Press play to watch the factors line up and collapse.

The same counting argument gives all three index laws. When you multiply powers of one base, add the indices:

When you divide, every factor on the bottom cancels one on top, so you subtract the indices:

And a power of a power means you have n copies of x^m, which is m factors repeated n times — so you multiply the indices:

Notice these only work when the base is the same letter. Powers of different bases — say x^2 \times y^3 — cannot be combined this way, just as you cannot merge unlike terms when collecting like terms.

Khan Academy works through multiplying powers of the same base here: