Multiplying Powers: Add the Exponents

≈ 25 min

When you multiply two powers with the same base, you may simply add the exponents:

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

Why does this work? Each power is just a pile of factors — count them:

2^3 \times 2^2 = (2 \times 2 \times 2) \times (2 \times 2) = 2^5

There are $3 + 2 = 5$ twos altogether. Adding the exponents simply counts the total factors.

Worked example. Simplify $x^4 \times x^3$ .

Same base $x$ , so add the exponents: $x^{4+3} = x^7$ .

Simplify $a^4 \times a^3$ .

$a^{12}$

$a^7$

$a^1$

$2a^7$

Write $3^2 \times 3^5$ as a single power, in the form base^exponent.

Evaluate $2^2 \times 2^3$ .

Simplify $x^5 \times x$ . Write your answer in the form x^n.