The Laws of Indices

Once a number is written as a power, we can combine powers without writing the products out in full — as long as they share the same base. The first two laws come straight from counting factors.

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

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

This is just counting copies of the base. For example 2^3 \times 2^2 is (2 \times 2 \times 2) \times (2 \times 2) — five twos altogether — so it is 2^{5}:

2^3 \times 2^2 = 2^{3+2} = 2^{5}

When you divide powers of the same base, you subtract the indices, because each factor on the bottom cancels one on the top:

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

5^6 \div 5^2 = 5^{6-2} = 5^{4}

The third law handles a power of a power. Raising a^m to the power n means writing n copies of a^m and multiplying — so by the first law the indices multiply:

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

For example (3^2)^3 is 3^2 \times 3^2 \times 3^2 = 3^{2+2+2}, which is 3^{2 \times 3} = 3^{6}.

For powers of the same base a:

Multiplying adds the indices — a^m \times a^n = a^{m+n}.
Dividing subtracts the indices — a^m \div a^n = a^{m-n}.
A power of a power multiplies the indices — (a^m)^n = a^{mn}.

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