# Lesson 3

Powers of Powers of 10

### Lesson Narrative

Students make use of repeated reasoning to discover the exponent rule $$\left(10^n\right)^m = 10^{n \boldcdot m}$$ (MP8). At this time, students develop rules for positive exponents. In subsequent lessons, students will extend the exponent rules to cases where the exponents are zero or negative. Students reason abstractly and quantitatively when applying exponent rules, pausing to consider the meaning of quantities, not just how to compute them (MP2).

### Learning Goals

Teacher Facing

• Generalize a process for finding a power raised to a power, and justify (orally and in writing) that $(10^n)^m = 10^{n \boldcdot m}$.

### Student Facing

Let's look at powers of powers of 10.

### Required Preparation

Create a visual display for the rule $$\left(10^m\right)^n = 10^{m\boldcdot n}$$ to display for all to see throughout the unit. The display will be introduced during the discussion of“Taking Powers of Powers of 10” activity. For an example of how the rule works, consider showing $$(10^2)^3 = (10 \boldcdot 10)(10 \boldcdot 10)(10 \boldcdot 10) = 10^6$$ using colors or other visual aids to highlight that the result is $$10^6$$ because there are three groups of $$10^2$$. For example,

### Student Facing

• I can explain and use a rule for raising a power of 10 to a power.

Building Towards

### 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 $$8^2 = 8 \boldcdot 8$$.