Product Rule for Exponents

The **product rule for exponents** says: when you multiply two powers with the **same base**, keep the base and **add the exponents**. $$a^m \cdot a^n = a^{m+n}$$ **Why it works: count the factors.** $2^3$ means three factors of $2$, and $2^4$ means four factors of $2$. Multiplying the two powers puts all the factors in one row: $$2^3 \cdot 2^4 = \underbrace{(2 \cdot 2 \cdot 2)}_{3 \text{ factors}} \cdot \underbrace{(2 \cdot 2 \cdot 2 \cdot 2)}_{4 \text{ factors}} = 2^{7}$$ Three factors and four factors make seven factors, so the exponents add: $3 + 4 = 7$. Check with numbers: $8 \cdot 16 = 128 = 2^7$. **Remember:** - The base **stays the same**: $2^3 \cdot 2^4 = 2^7$, never $4^7$. - The exponents are **added**, not multiplied: $2^7$, not $2^{12}$. - The bases must be **equal**: $2^3 \cdot 5^2$ cannot be written as one power. - A base with no written exponent has the exponent $1$: $x = x^1$.

Same Base, Add the Exponents

Simplify $3^2 \cdot 3^5$ and check the result.

Check the bases: Both powers have the base $3$ = Same base, so the product rule applies

Add the exponents: $3^2 \cdot 3^5 = 3^{2+5}$ = $3^7$

Check by counting factors: $(3 \cdot 3) \cdot (3 \cdot 3 \cdot 3 \cdot 3 \cdot 3)$ has $2 + 5 = 7$ factors of $3$ = $3^7$ ✓

Check with numbers: $3^2 = 9$, $3^5 = 243$, $9 \cdot 243 = 2187$ = $3^7 = 2187$ ✓

Answer: $3^2 \cdot 3^5 = 3^7 = 2187$

Powers of a Variable

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

Write out the factors: $x^2 = x \cdot x$ and $x^5 = x \cdot x \cdot x \cdot x \cdot x$ = 2 factors and 5 factors

Count all the factors of $x$: $2 + 5 = 7$ = 7 factors of $x$

Write the result as one power: $x^2 \cdot x^5 = x^{2+5}$ = $x^7$

Answer: $x^2 \cdot x^5 = x^7$

The Hidden Exponent 1

Simplify $5 \cdot 5^3 \cdot 5^2$.

Give every factor an exponent: $5 = 5^1$ = $5^1 \cdot 5^3 \cdot 5^2$

Add all three exponents: $1 + 3 + 2 = 6$ = $5^6$

Find the value: $5^6 = 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$ = $15\,625$

Answer: $5 \cdot 5^3 \cdot 5^2 = 5^6 = 15\,625$

Coefficients and Powers

Simplify $3x^2 \cdot 4x^3$.

Reorder the factors (you may multiply in any order): $3x^2 \cdot 4x^3 = (3 \cdot 4) \cdot (x^2 \cdot x^3)$ = Numbers together, powers together

Multiply the coefficients: $3 \cdot 4 = 12$ = $12$

Add the exponents of $x$: $x^2 \cdot x^3 = x^{2+3}$ = $x^5$

Put the parts together: $12 \cdot x^5$ = $12x^5$

Answer: $3x^2 \cdot 4x^3 = 12x^5$

Two Different Variables

Simplify $2a^3b \cdot 5a^4b^2$.

Group the numbers and each variable: $(2 \cdot 5) \cdot (a^3 \cdot a^4) \cdot (b \cdot b^2)$ = Three groups

Multiply the coefficients: $2 \cdot 5 = 10$ = $10$

Base $a$: add the exponents: $a^3 \cdot a^4 = a^{3+4}$ = $a^7$

Base $b$: remember that $b = b^1$: $b^1 \cdot b^2 = b^{1+2}$ = $b^3$

Write the result: $a$ and $b$ are different bases, so they stay separate = $10a^7b^3$

Answer: $2a^3b \cdot 5a^4b^2 = 10a^7b^3$

The product rule saves you from writing out long rows of factors, and it shows up all over math and science: - **Scientific notation**: $10^3 \cdot 10^4 = 10^7$, so a thousand times ten thousand is ten million. - **Computer memory**: $2^{10}$ bytes make a kilobyte and $2^{10}$ kilobytes make a megabyte, so a megabyte is $2^{10} \cdot 2^{10} = 2^{20}$ bytes. - **Repeated doubling**: something that doubles 3 times and then 4 more times has grown $2^3 \cdot 2^4 = 2^7 = 128$ times. - **Algebra**: areas and volumes like $x^2 \cdot x$ or $3x^2 \cdot 4x^3$ must be simplified before you can solve equations or work with polynomials. It is also the rule the others are built on: the quotient rule and the power of a power follow from counting factors in exactly the same way.

Mistake: Multiplying the exponents: $2^3 \cdot 2^4 = 2^{12}$

Why: The exponents count factors. Three factors and four factors are $3 + 4 = 7$ factors, not $3 \cdot 4 = 12$. Check: $8 \cdot 16 = 128$, but $2^{12} = 4096$. Multiplying exponents belongs to a power of a power, like $(2^3)^4$.

Correct: $2^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128$

Mistake: Multiplying the bases: $2^3 \cdot 2^4 = 4^7$

Why: Every factor in the row is still a $2$. Nothing turns into a $4$. And $4^7 = 16\,384$, far more than the real product $128$.

Correct: Keep the base: $2^3 \cdot 2^4 = 2^7$

Mistake: Adding exponents of different bases: $2^3 \cdot 3^2 = 6^5$ or $5^5$

Why: The product rule needs the same base, because only then do all the factors match. $2^3 \cdot 3^2 = 8 \cdot 9 = 72$, while $6^5 = 7776$.

Correct: Leave it as $2^3 \cdot 3^2$ or calculate it: $8 \cdot 9 = 72$

Mistake: Adding the coefficients: $3x^2 \cdot 4x^3 = 7x^5$

Why: The coefficients $3$ and $4$ are ordinary factors, so they are multiplied. Only the exponents of the same base are added.

Correct: $3x^2 \cdot 4x^3 = (3 \cdot 4)\,x^{2+3} = 12x^5$

Computer Memory

Computers count memory in powers of 2, so memory sizes are multiplied with the product rule.

A kilobyte is $2^{10}$ bytes and a megabyte is $2^{10}$ kilobytes, so a megabyte is $2^{10} \cdot 2^{10} = 2^{20}$ bytes.

Counting in Powers of Ten

Large counts in shops and warehouses are easy to handle as powers of 10.

$10^2$ boxes with $10^3$ screws each hold $10^2 \cdot 10^3 = 10^5 = 100\,000$ screws.

Folding Paper

Every fold doubles the layers of a sheet of paper, so the number of layers grows in powers of 2.

After 2 folds there are $2^2 = 4$ layers, after 3 folds there are $2^3 = 8$ layers.

**Product rule**: $a^m \cdot a^n = a^{m+n}$. Same base, add the exponents.

It works because you count factors: $2^3 \cdot 2^4$ has $3 + 4 = 7$ factors of $2$.

The base stays the same: $2^3 \cdot 2^4 = 2^7$, not $4^7$.

The exponents are added, not multiplied: $2^3 \cdot 2^4 = 2^7$, not $2^{12}$.

A base with no written exponent has the exponent 1: $x \cdot x^4 = x^5$.

With coefficients, multiply the numbers and add the exponents of each base: $3x^2 \cdot 4x^3 = 12x^5$. Different bases stay separate.

Q: Why do we add the exponents and not multiply them?

A: Each exponent counts factors. $2^3$ is 3 factors of 2 and $2^4$ is 4 more factors of 2, so together there are $3 + 4 = 7$ factors: $2^7$. Multiplying exponents belongs to a different situation, a power raised to a power like $(2^3)^4$, which has its own lesson.

Q: Does the rule work for more than two powers?

A: Yes. Add all the exponents: $2 \cdot 2^2 \cdot 2^3 = 2^{1+2+3} = 2^6 = 64$.

Q: Can I use the product rule for $2^3 \cdot 3^3$?

A: No, the bases are different, so you cannot add the exponents. Here the exponents happen to be equal, and then a different rule helps: $2^3 \cdot 3^3 = (2 \cdot 3)^3 = 6^3 = 216$. Check: $8 \cdot 27 = 216$. With different bases and different exponents, like $2^3 \cdot 3^2$, just calculate: $8 \cdot 9 = 72$.

Q: Is $2^3 + 2^4$ equal to $2^7$?

A: No. The product rule is only for multiplication. $2^3 + 2^4 = 8 + 16 = 24$, while $2^7 = 128$.

Exit Presentation

Product Rule for Exponents

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Arithmetic / Exponents & Roots

Product Rule for Exponents

Learn why multiplying powers with the same base means adding the exponents, with numbers, variables and coefficients.

Intermediate20 min

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