What Are Exponent Laws?

When we write $a^{n}$ , the number $a$ is called the base and $n$ is called the exponent (or power or index). The exponent tells us how many times the base is multiplied by itself.

Exponent (Power/Index)

If $n$ is a positive integer, then $a^{n}$ means $a$ multiplied by itself $n$ times: $a^{n} = a \times a \times a \times \dots \times a (n factors)$

As expressions involving exponents get more complex, it becomes very useful to have laws (rules) that let us simplify them without expanding everything out. There are six key exponent laws you need to master at MYP 3.

Note

All the laws in this section work when the bases are the same non-zero number. Always check the bases match before applying a rule!

Law 1 — Multiplying Powers (Product Rule)

Consider $3^{2} \times 3^{4}$ . If we expand:
$3^{2} \times 3^{4} = (3 \times 3) \times (3 \times 3 \times 3 \times 3) = 3^{6}$
Notice that $2 + 4 = 6$ . The exponents are added.

Product Rule

If $m$ and $n$ are integers and $a \neq = 0$ , then: $a^{m} \times a^{n} = a^{m + n}$ To multiply powers with the same base, keep the base and add the exponents.

Analogy

Think of it like counting apples in two bags. If one bag has 2 groups-of-3 and the other has 4 groups-of-3, you just count all the groups together: $2 + 4 = 6$ groups of 3.

Example

Simplify the following using the product rule.

(a) $2^{3} \times 2^{5}$
$= 2^{3 + 5} = 2^{8}$

(b) $x^{4} \times x^{6}$
$= x^{4 + 6} = x^{10}$

(c) $a^{3} \times a$
$= a^{3} \times a^{1} = a^{3 + 1} = a^{4}$
(Remember: if no exponent is written, it is 1!)

Common Mistake

Don't multiply the bases! $3^{2} \times 3^{4} \neq = 9^{6}$ . The base stays the same — only the exponents are added.

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