## 1.1: Which One Doesn’t Belong: Twos

Which expression does not belong? Be prepared to share your reasoning.

$2^3$

8

$3^2$

$2^2\boldcdot 2^1$

Let’s review exponents.

Which expression does not belong? Be prepared to share your reasoning.

$2^3$

8

$3^2$

$2^2\boldcdot 2^1$

Mai and Andre found an old, brass bottle that contained a magical genie. They freed the genie, and it offered them each a magical \$1 coin as thanks. The magic coin turned into 2 coins on the first day. The 2 coins turned into 4 coins on the second day. On the third day, the 4 coins magically turned into 8 coins, and this pattern continued for 28 days.

Click on the arrow to see the coins start to magically multiply.

Mai was trying to calculate how many coins she would have and remembered that instead of writing $1 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2$ for the number of coins on the 6th day, she could write $2^6$.

- The number of coins Mai had on the 28th day is very, very large. Write an expression to represent this number without computing its value.
- Andre’s coins lost their magic on the 25th day, so Mai has a lot more coins than he does. How many times more coins does Mai have than Andre?

After a while, Jada picks up a coin that seems different than the others. She notices that the next day, only half of the coin is left! On the second day, only $\frac{1}{4}$ of the coin is left. On the third day, $\frac{1}{8}$ of the coin remains.

- What fraction of the coin remains after 6 days?
- What fraction of the coin remains after 28 days? Write an expression to describe this without computing its value.
- Does the coin disappear completely? If so, after how many days?

Watch the magical coin changing for ten days with this applet.

Tyler has two parents. Each of his parents also has two parents.

- Draw a family tree showing Tyler, his parents, his grandparents, and his great-grandparents.
- We say that Tyler’s eight great-grandparents are “three generations back” from Tyler. At which generation back does Tyler have 262,144 ancestors?

Exponents make it easy to show repeated multiplication. For example, $$2^6 = 2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2.$$One advantage to writing $2^6$ is that we can see right away that this is 2 to the *sixth* power. When this is written out using multiplication, $2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2$, we need to count the number of factors. Imagine writing out $2^{100}$ using multiplication!

Let’s say you start out with one grain of rice and that each day the number of grains of rice you have doubles. So on day one, you have 2 grains, on day two, you have 4 grains, and so on. When we write $2^{25}$, we can see from the expression that the rice has doubled 25 times. So this notation is not only convenient, but it also helps us see structure: in this case, we can see right away that it is on the 25th day that the number of grains of rice has doubled! That’s a lot of rice (more than a cubic meter)!