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 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


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 \(\frac52\).