# Lesson 18

Bases and Exponents

• Let’s rewrite expressions using the property $$(x^a)^b = x^{ab}$$.

### 18.1: Math Talk: Different Bases

Decide if each expression is equal to $$9^{16}$$.

$$(9^{8})^{8}$$

$$(9^4)^4$$

$$(3^2)^{16}$$

$$3^{32}$$

### 18.2: What’s the Factor?

1. Refer to the first table.
 step value expression 0 1 2 3 4 5 6 10 30 90 270 $$10\boldcdot3^0$$ $$10\boldcdot3^1$$ $$10\boldcdot3^2$$
1. Predict the value in steps 4, 5, and 6.
2. By what factor does the value change . . .
1. from step 1 to step 4?
2. from step 3 to step 6?
3. Conjecture about the factor from step 7 to step 10.
3. By what factor does the value change . . .
1. from step 0 to step 5?
2. from step 1 to step 6?
3. Conjecture about the factor from step 10 to step 15.
2. Refer to the second table.
 step value expression 0 1 2 3 4 5 6 3 6 12 24 $$3\boldcdot2^0$$
1. Predict the value in steps 4, 5, and 6.
2. By what factor does the value change . . .
1. from step 1 to step 3?
2. from step 3 to step 5?
3. Conjecture about the factor from step 10 to step 12.
3. By what factor does the value change . . .
1. from step 0 to step 3?
2. from step 2 to step 5?
3. Conjecture about the factor from step 10 to step 13.
3. Refer to the third table.
 step value expression 0 1 2 3 4 5 6 2,048 1,024 512
1. Predict the value in steps 4, 5, and 6.
2. By what factor does the value change . . .
1. from step 1 to step 3?
2. from step 3 to step 5?
3. Conjecture about the factor from step 10 to step 12.
3. By what factor does the value change . . .
1. from step 0 to step 3?
2. from step 2 to step 5?
3. Conjecture about the factor from step 10 to step 13.

### 18.3: Rewriting Expressions

1. For each given expression, decide what to write in the box to create equal expressions.
given expression equal expression 1 equal expression 2
$$5\boldcdot10^8$$ $$5\boldcdot100^\boxed{\phantom{3}}$$ $$5\boldcdot\boxed{\phantom{3}}^2$$
$$7\boldcdot16^9$$ $$7\boldcdot\boxed{\phantom{3}}^{4\boldcdot9}$$ $$7\boldcdot4^\boxed{\phantom{3}}$$
$$(0.25)^3$$ $$(0.5)^\boxed{\phantom{3}}$$ $$\boxed{\phantom{3}}^1$$
$$3\boldcdot(1.2)^6$$ $$3\boldcdot1.44^\boxed{\phantom{3}}$$ $$3\boldcdot1.728^\boxed{\phantom{3}}$$
$$6\boldcdot0.09^{10}$$ $$6\boldcdot\boxed{\phantom{3}}^5$$ $$6\boldcdot0.3^\boxed{\phantom{3}}$$

2. Write at least 3 new expressions that are equal to $$4 \boldcdot 27^6$$.