Lesson 8: Combining Bases

Let’s multiply expressions with different bases.

8.1: Same Exponent, Different Base

  1. Evaluate $5^3 \boldcdot 2^3$
  2. Evaluate $10^3$

8.2: Exponent Product Rule

  1. The table contains products of expressions with different bases and the same exponent. Complete the table to see how we can rewrite them. Use the “expanded” column to work out how to combine the factors into a new base.

    Row 1 expression expanded exponent
    Row 2 $5^3 \boldcdot 2^3$ $\begin{align}(5 \boldcdot 5 \boldcdot 5) \boldcdot (2 \boldcdot 2 \boldcdot 2) &= (2 \boldcdot 5)(2 \boldcdot 5)(2 \boldcdot 5)\\ &= 10 \boldcdot 10 \boldcdot 10 \end{align}$ $10^3$
    Row 3 $3^2 \boldcdot 7^2$   $21^2$
    Row 4 $2^4 \boldcdot 3^4$    
    Row 5     $15^3$
    Row 6     $30^4$
    Row 7 $2^4 \boldcdot x^4$    
    Row 8 $a^n \boldcdot b^n$    
    Row 9 $7^4 \boldcdot 2^4 \boldcdot 5^4$    
  2. What happens if neither the exponents nor the bases are the same? Can you write $2^3 \boldcdot 3^4$ with a single exponent? Explain or show your reasoning.

8.3: How Many Ways Can You Make 3,600?

Your teacher will give your group tools for creating a visual display to play a game. Divide the display into 3 columns, with these headers:

$a^n \boldcdot a^m = a^{n+m}$

$\frac{a^n}{a^m} = a^{n-m}$

$a^n \boldcdot b^n = (a \boldcdot b)^n$

How to play:

When the time starts, you and your group will write as many expressions as you can that equal a specific number using one of the exponent rules on your board. When the time is up, compare your expressions with another group to see how many points you earn.

  • Your group gets 1 point for every unique expression you write that is equal to the number and follows the exponent rule you claimed.
  • If an expression uses negative exponents, you get 2 points instead of just 1.
  • You can challenge the other group’s expression if you think it is not equal to the number or if it does not follow one of the three exponent rules.


Before this lesson, we made rules for multiplying and dividing expressions with exponents that only work when the expressions have the same base. For example, $$10^3 \boldcdot 10^2 = 10^5$$ or $$2^6 \div 2^2 = 2^4$$ 

In this lesson, we studied how to combine expressions with the same exponent, but different bases. For example, we can write $2^3 \boldcdot 5^3$ as $2 \boldcdot 2 \boldcdot 2 \boldcdot 5 \boldcdot 5 \boldcdot 5$. Regrouping this as $(2 \boldcdot 5) \boldcdot (2 \boldcdot 5) \boldcdot (2 \boldcdot 5)$ shows that

$$\begin{align}2^3 \boldcdot 5^3 &= (2 \boldcdot 5)^3\\ & = 10^3 \end{align}$$

Notice that the 2 and 5 in the previous example could be replaced with different numbers or even variables. For example, if $a$ and $b$ are variables then $a^3 \boldcdot b^3 = (a \boldcdot b)^3$. More generally, for a positive number $n$, $$a^n \boldcdot b^n = (a \boldcdot b)^n$$ because both sides have exactly $n$ factors that are $a$ and $n$ factors that are $b$.

Practice Problems ▶