Combining Bases

5 min

Teacher Prep
Setup
Give students 2 minutes of quiet work time followed by a whole-class discussion.

Narrative

The purpose of this Warm-Up is to encourage students to relate expressions of the form an⋅bna^n \boldcdot b^n to (a⋅b)n(a \boldcdot b)^n by exploring the structure of the factors (MP7). Evaluating and expanding expressions will be useful when students explore products of bases with the same exponent in a following activity.

Launch

Give students 2 minutes of quiet work time followed by a whole-class discussion. 

Student Task

  1. Evaluate 53⋅235^3 \boldcdot 2^3.
  2. Evaluate 10310^3.

Sample Response

  1. 53⋅23=125⋅8=1,0005^3 \boldcdot 2^3 = 125 \boldcdot 8 = 1,000.
  2. 103=1,00010^3 = 1,000.
Activity Synthesis (Teacher Notes)

The purpose of this discussion is to help students make connections between the two expressions. Here are some questions for discussion:

  • “What do you notice about the two expressions?” (The product of the bases in the first expression is equal to the base in the second expression: 2⋅5=102 \boldcdot 5 = 10. The exponents are the same in both expressions.)
  • “How can we show that the two expressions are equivalent without evaluating?” (Since there are 3 factors that are 5 and 3 factors that are 2, group the 2s and 5s together to get 3 factors that are 10.
    53⋅23=(5⋅5⋅5)⋅(2⋅2⋅2)=(5⋅2)⋅(5⋅2)⋅(5⋅2)=10⋅10⋅10=1035^3\boldcdot2^3=(5\boldcdot5\boldcdot5)\boldcdot(2\boldcdot2\boldcdot2)=(5\boldcdot2)\boldcdot(5\boldcdot2)\boldcdot(5\boldcdot2)=10\boldcdot10\boldcdot10=10^3.)
Standards
Addressing
  • 8.EE.1·Know and apply the properties of integer exponents to generate equivalent numerical expressions. <em>For example, 3² × 3<sup>-5</sup> = 3<sup>-3</sup> = 1/3³ = 1/27.</em>
  • 8.EE.A.1·Know and apply the properties of integer exponents to generate equivalent numerical expressions. <span>For example, <span class="math">\(3^2\times3^{-5} = 3^{-3} = 1/3^3 = 1/27\)</span>.</span>

20 min