8.1: Same Exponent, Different Base
- Evaluate $5^3 \boldcdot 2^3$
- Evaluate $10^3$
Let’s multiply expressions with different bases.
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 | $7^4 \boldcdot 2^4 \boldcdot 5^4$ |
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.
You have probably noticed that when you square an odd number, you get another odd number, and when you square an even number, you get another even number. Here is a way to expand the concept of odd and even for the number 3. Every integer is either divisible by 3, one MORE than a multiple of 3, or one LESS than a multiple of 3.
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 $b$.