Lesson 8
Combining Bases
Lesson Narrative
Previously, students saw that the exponent rules thus far only apply when the bases are the same. In this lesson, students explore what happens when bases are different. This leads to the rule \(a^n b^n = (a \boldcdot b)^n\). Students make use of structure when decomposing numbers into their constituent factors and regrouping them (MP7). Students create viable arguments and critique the reasoning of others when they generate expressions equivalent to 3,600 and \(\frac{1}{200}\) using exponent rules and determine the validity of other teams’ expressions (MP3).
Learning Goals
Teacher Facing
 Generalize a process for multiplying expressions with different bases having the same exponent, and justify (orally and in writing) that $(ab)^n = a^n \boldcdot b^n$.
Student Facing
Let’s multiply expressions with different bases.
Required Materials
Required Preparation
Create a visual display for the rule \((a \boldcdot b)^n = a^n \boldcdot b^n\). As a guiding example, consider \(2^3 \boldcdot 5^3 = 2 \boldcdot 2 \boldcdot 2 \boldcdot 5 \boldcdot 5 \boldcdot 5 = (2\boldcdot 5)\boldcdot (2\boldcdot 5)\boldcdot (2\boldcdot 5) = 10 \boldcdot 10 \boldcdot 10 = 10^3\).
Learning Targets
Student Facing
 I can use and explain a rule for multiplying terms that have different bases but the same exponent.
CCSS Standards
Addressing
Glossary Entries

base (of an exponent)
In expressions like \(5^3\) and \(8^2\), the 5 and the 8 are called bases. They tell you what factor to multiply repeatedly. For example, \(5^3\) = \(5 \boldcdot 5 \boldcdot 5\), and \(8^2 = 8 \boldcdot 8\).

reciprocal
Dividing 1 by a number gives the reciprocal of that number. For example, the reciprocal of 12 is \(\frac{1}{12}\), and the reciprocal of \(\frac25\) is \(\frac52\).