# Lesson 8

Combining Bases

### Problem 1

Select all the true statements:

A:

$$2^8 \boldcdot 2^9 = 2^{17}$$

B:

$$8^2 \boldcdot 9^2 = 72^2$$

C:

$$8^2 \boldcdot 9^2 = 72^4$$

D:

$$2^8 \boldcdot 2^9 = 4^{17}$$

### Problem 2

Find $$x$$, $$y$$, and $$z$$ if $$(3 \boldcdot 5)^4 \boldcdot (2 \boldcdot 3)^5 \boldcdot (2 \boldcdot 5)^7 = 2^x \boldcdot 3^y \boldcdot 5^z$$.

### Problem 3

Han found a way to compute complicated expressions more easily. Since $$2 \boldcdot 5 = 10$$, he looks for pairings of 2s and 5s that he knows equal 10. For example, $$3 \boldcdot 2^4 \boldcdot 5^5 = 3 \boldcdot 2^4 \boldcdot 5^4 \boldcdot 5 = (3 \boldcdot 5) \boldcdot (2 \boldcdot 5)^4 = 15 \boldcdot 10^4 = 150,\!000.$$ Use Han's technique to compute the following:

1. $$2^4 \boldcdot 5 \boldcdot (3 \boldcdot 5)^3$$
2. $$\frac{2^3 \boldcdot 5^2 \boldcdot (2 \boldcdot 3)^2 \boldcdot (3 \boldcdot 5)^2}{3^2}$$

### Problem 4

The cost of cheese at three stores is a function of the weight of the cheese. The cheese is not prepackaged, so a customer can buy any amount of cheese.

• Store A sells the cheese for $$a$$ dollars per pound.

• Store B sells the same cheese for $$b$$ dollars per pound and a customer has a coupon for $5 off the total purchase at that store. • Store C is an online store, selling the same cheese at $$c$$ dollar per pound, but with a$10 delivery fee.

This graph shows the price functions for stores A, B, and C.

1. Match Stores A, B, and C with Graphs $$j$$, $$k$$, and $$\ell$$.

2. How much does each store charge for the cheese per pound?

3. How many pounds of cheese does the coupon for Store B pay for?

4. Which store has the lowest price for a half a pound of cheese?

5. If a customer wants to buy 5 pounds of cheese for a party, which store has the lowest price?

6. How many pounds would a customer need to order to make Store C a good option?