Lesson 8

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).

Create a visual display for the rule \((a \boldcdot b)^n = a^n \boldcdot b^n\). As a guiding example, consider \(2^3 \boldcdot 5^3 = 2 \boldcdot 2 \boldcdot 2 \boldcdot 5 \boldcdot 5 \boldcdot 5 = (2\boldcdot 5)\boldcdot (2\boldcdot 5)\boldcdot (2\boldcdot 5) = 10 \boldcdot 10 \boldcdot 10 = 10^3\).