Multiplication Rule

Q: What's the exponent on a plain variable like $x$?

It's $1$. So $x \cdot x^{6} = x^{1+6} = x^{7}$. Forgetting that invisible $1$ is one of the most common errors on these problems.

When you multiply two powers that have the same base, you keep the base and add the exponents: $x^{5} \cdot x^{2} = x^{7}$ . That's the multiplication rule (also called the product rule), and it's the first exponent rule everything else builds on.

The rule isn't magic — it's just counting. $x^{5}$ means five $x$ 's multiplied together, and $x^{2}$ means two more. Multiply them and you have seven $x$ 's in a row, which is exactly $x^{7}$ . Once you see it as counting factors, you'll never mix it up with the other rules.

The rule

Same base, multiplied: keep the base, add the exponents. In symbols, $x^{m} \cdot x^{n} = x^{m+n}$ .

The base has to match. $x^{3} \cdot x^{4}$ becomes $x^{7}$ , but $x^{3} \cdot y^{4}$ can't be combined — those are different bases, so you just leave them side by side.

One detail that trips people up: a plain variable like $x$ has an invisible exponent of $1$ . So $x^{4} \cdot x$ is really $x^{4} \cdot x^{1} = x^{5}$ .

What about the numbers in front?

Coefficients — the numbers attached to the variables — get multiplied normally. Only the exponents get added. In $4x^{3} \cdot 2x^{5}$ , multiply $4 \cdot 2 = 8$ and add $3 + 5 = 8$ , giving $8x^{8}$ .

With two variables, handle each one separately: in $(2x^{3}y^{2})(3xy^{4})$ , the coefficients give $6$ , the $x$ exponents give $3 + 1 = 4$ , and the $y$ exponents give $2 + 4 = 6$ . Result: $6x^{4}y^{6}$ .

The classic mistake

Students often multiply the exponents instead of adding them, writing $x^{5} \cdot x^{2} = x^{10}$ . Check it with small numbers: $2^{2} \cdot 2^{3} = 4 \cdot 8 = 32 = 2^{5}$ , not $2^{6} = 64$ . Multiplying powers adds exponents; raising a power to a power is what multiplies them.

Worked examples

Example 1: same base, add the exponents

Simplify $x^{5} \cdot x^{2}$ .

Same base

x

, so keep it and add the exponents

x^{5+2}

Add

x^{7}

Answer: $x^{7}$

Example 2: coefficients multiply, exponents add

Simplify $4x^{3} \cdot 2x^{5}$ .

Multiply the coefficients

4 \cdot 2 = 8

Add the exponents on

x

3 + 5 = 8

Put them together

8x^{8}

Answer: $8x^{8}$

Example 3: two variables

Simplify $(2x^{3}y^{2})(3xy^{4})$ .

Multiply the coefficients

2 \cdot 3 = 6

Add the exponents on

x

(remember

x = x^{1}

)

3 + 1 = 4

Add the exponents on

y

2 + 4 = 6

Put them together

6x^{4}y^{6}

Answer: $6x^{4}y^{6}$

Try one yourself

Common questions

Why do I add the exponents instead of multiplying them?

Because exponents count factors. $x^{5} \cdot x^{2}$ is five $x$ 's times two $x$ 's — seven $x$ 's total. Adding the exponents is just adding the counts.

Can I use the rule when the bases are different?

No. $x^{3} \cdot y^{4}$ stays as $x^{3}y^{4}$ — there's nothing to combine. The rule only applies when the bases match exactly.

What's the exponent on a plain variable like

x

It's $1$ . So $x \cdot x^{6} = x^{1+6} = x^{7}$ . Forgetting that invisible $1$ is one of the most common errors on these problems.

Do the coefficients follow the same rule?

No — coefficients multiply the normal way. In $5x^{2} \cdot 3x^{4}$ , the $5$ and $3$ multiply to $15$ , while the exponents add to $6$ , giving $15x^{6}$ .

Want the video version?

Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.

Try it for $1

Multiplication Rule

The rule

What about the numbers in front?

The classic mistake

Worked examples

Example 1: same base, add the exponents

Example 2: coefficients multiply, exponents add

Example 3: two variables

Try one yourself

Common questions

Keep learning

Division Rule

Power Rule

Negative Exponent Rule