# Lesson 4

Dividing Powers of 10

### Lesson Narrative

Students continue to use repeated reasoning to discover the exponent rule $$\frac{10^n}{10^m} = 10^{n-m}$$ (MP8). For now, students work with expressions where $$n$$ and $$m$$ are positive integers and $$n > m$$. In the last activity, students extend to the case where $$n = m$$ to make sense of why $$10^0$$ is defined to be equal to 1 and critique a faulty argument that it should be defined to be equal to 0 (MP3). Students make sense of this rule when they recognize that separating the same number of factors from the numerator and denominator, then dividing has the effect of multiplying by 1. This essentially means that $$m$$ factors are subtracted from the $$n$$ factors in the numerator. For example $$\frac{10^3}{10^2} = \frac{10 \boldcdot 10}{10 \boldcdot 10} \boldcdot 10$$ which is the same as $$1 \boldcdot 10 = 10$$ or the same as $$10^{3-2} = 10^1$$. In a subsequent lesson, students will extend this rule to include situations where $$n < m$$.

### Learning Goals

Teacher Facing

• Generalize a process for dividing powers of 10, and justify (orally and in writing) that $\frac{10^n}{10^m} = 10^{n−m}$.
• Use exponent rules to multiply and divide with $10^0$, and justify (orally) that $10^0$ is 1.

### Student Facing

Let’s explore patterns with exponents when we divide powers of 10.

### Required Preparation

Create a visual display for the rule $$\frac{10^n}{10^m} = 10^{n-m}$$. For a guiding example, consider $$\frac{10^5}{10^2} = \frac{10 \boldcdot 10 \boldcdot 10 \boldcdot 10 \boldcdot 10}{10 \boldcdot 10} = \frac{10 \boldcdot 10}{10 \boldcdot 10} \boldcdot 10 \boldcdot 10 \boldcdot 10 = 1 \boldcdot 10^3 = 10^3$$.

Create a visual display for the rule $$10^0 = 1$$. For an example, you can show $$\frac{10^6}{1}=\frac{10^6}{10^0}=10^{6−0}=10^6$$. Another possibility is to write $$10^5\boldcdot 1= 10^5\boldcdot 10^0=10^{5+0}=10^5$$ and use visual aids to highlight that each of these examples implies $$10^0=1$$.

### Student Facing

• I can evaluate $10^0$ and explain why it makes sense.
• I can explain and use a rule for dividing powers of 10.

Building On