You look at x² · x³ and the instinct fires: two exponents, there's a multiplication sign between them, so you multiply the exponents. Answer: x⁶.

Wrong. By a factor of x.

The correct answer is x⁵. When you multiply powers that share the same base, you add the exponents, you do not multiply them. x² · x³ = x^(2+3) = x⁵. That rule has a reason, and once you see the reason, you will never mix it up with the power-of-a-power rule again.

Test with numbers

Before proving anything, test it. Set x = 2 and see which answer is right.

The x⁶ answer is not slightly off. It is off by a whole factor of x. If x = 10, the mistake turns 10⁵ = 100000 into 10⁶ = 1000000 — a wrong answer ten times too big.

Numbers catch the error immediately. If you are ever unsure in an exam, plug in x = 2 or x = 3 and check.

The product rule, proved

Forget rules for a moment. What does x² actually mean? It means x · x. Two copies of x, multiplied together. And x³ means x · x · x. Three copies.

Now write out x² · x³ by expanding both:

x² · x³ = (x · x) · (x · x · x)

Multiplication is associative, so the brackets do not matter. Remove them:

= x · x · x · x · x = x⁵

Count the x's. Two from the first factor, three from the second, total five. The number of x's adds: 2 + 3 = 5. It does not multiply: 2 × 3 = 6 would require six copies of x, and there are only five.

That is the entire story. When you multiply same-base powers, you are concatenating two lists of x's into one longer list, and lengths of lists add.

Where "multiply exponents" comes from — the power-of-a-power rule

The multiplication-of-exponents instinct is not random. There is a real rule where you multiply exponents. It just applies to a different situation.

Consider (x²)³. This means "take x², and cube it":

(x²)³ = x² · x² · x² = (x·x)(x·x)(x·x) = x⁶

Three copies of x², each containing two x's, makes six x's total. Here the 3 does multiply the 2, because you have three instances of "two copies of x". That is 2 × 3 = 6 copies.

This is a different operation. (x²)³ is a power of a power — an exponentiated thing, raised to another exponent. x² · x³ is a product of powers — two powers multiplied by each other.

The confusion is verbal. Both situations involve the word "multiply" somewhere: one multiplies two powers, the other multiplies an exponent by an exponent. Your brain latches onto "multiply" and produces the wrong output.

The decision rule

Two rules, clearly stated:

Before applying a rule, read the structure. Is the outside operation a multiplication sign between two separate powers? Add. Is the outside operation an exponent wrapping a parenthesised power? Multiply.

Ask: is there a · between two bases, or is there a (...)^something structure? The answer tells you which rule.

Worked examples of the product rule

Add exponents, keep the base.

Worked examples of the power-of-a-power rule

Multiply exponents.

Contrasting them side-by-side

Put the two situations next to each other with the same numbers:

x² · x³ = x⁵ (add: 2+3=5)

(x²)³ = x⁶ (multiply: 2·3=6)

Same base x, same small exponents 2 and 3, but the answers differ by a factor of x. This is exactly the distinction students miss. The difference is the bracketing and the position of the second exponent — is it the exponent of another x, or is it the exponent wrapping the whole x²?

Including coefficients

Real monomials come with coefficients — numbers in front. The rule extends naturally: multiply the coefficients, add the exponents.

3x² · 5x³ = (3 · 5) · x^(2+3) = 15x⁵

Not 15x⁶. The coefficients multiply because they really are just numbers being multiplied (3 and 5 give 15). The exponents add because the x-parts follow the product-of-powers rule.

Writing 15x⁶ mixes the coefficient rule (multiplication is correct) with the wrong rule for the exponents.

When there are multiple variables

Different variables are independent. Apply the product rule to each one separately.

3x²y · 5xy³ = (3 · 5) · x^(2+1) · y^(1+3) = 15x³y⁴

The x-exponents add (2 and 1 make 3). The y-exponents add independently (1 and 3 make 4). Coefficients multiply once. The x's and y's do not interact; they each have their own exponent accountant.

Recognition drill

State the correct answer for each. Recognise which rule applies first, then apply it.

That last one catches people. When you raise a whole monomial to a power, every factor inside gets raised to that power — coefficient included.

The "exponent-addition vs multiplication" memory trick

One line to hold in your head:

The operator before the exponent tells you what to do with the exponent.

Or even shorter: · on the outside means + on the exponents. ^ on the outside means · on the exponents. The outside operation drops one level in the operation hierarchy when it reaches the exponents.

Why memorising the rule without the reason fails

If you memorise "x² · x³ = x⁵" and "(x²)³ = x⁶" as two isolated facts, you will mix them up under exam pressure. The surface features — two exponents, a multiplication sign somewhere — look identical.

But if you remember the concatenation argument — that x² just means two x's stacked, and multiplying x² · x³ concatenates those stacks — both rules become obvious. Count the x's. That is all.

x² · x³: two x's then three x's, stacked side by side, makes five.

(x²)³: three copies of "two x's", stacked, makes six.

You never actually need the rules once you think this way. You reconstruct them every time.

Closing

Multiplying powers with the same base adds the exponents. Raising a power to a power multiplies the exponents. Two different operations, two different rules. If you ever freeze mid-problem, expand and count the x's; the right exponent falls out of the counting without you needing to remember which rule is which.