## Problem 1

1. ${\displaystyle 7^{3}}$
2. ${\displaystyle 5+4^{2}}$
3. ${\displaystyle 1213-9^{3}}$

${\displaystyle 7^{3}}$ ${\displaystyle 5+4^{2}}$ ${\displaystyle 1213-9^{3}}$
${\displaystyle 7\times 7\times 7}$ ${\displaystyle 5+4\times 4}$ ${\displaystyle 1213-9\times 9\times 9}$
${\displaystyle 49\times 7}$ Multiply the 4s first Multiply the 9s first
343 ${\displaystyle 5+16}$ ${\displaystyle 1213-729}$
21 484

## Problem 2

2.a ${\displaystyle 10^{4}}$
2.b ${\displaystyle 10^{7}}$
2.c ${\displaystyle 10^{1}0}$

 2.a ${\displaystyle 10^{4}=10\times 10\times 10\times 10=10,000}$ 2.b ${\displaystyle 10^{7}=10\times 10\times 10\times 10\times 10\times 10\times 10=10,000,000}$  2.c ${\displaystyle 10^{1}0=10,000,000,000}$


## Problem 3

Everybody is born to ${\displaystyle 2^{1}}$ biological parents. Our parents each had ${\displaystyle 2^{1}+2^{1}}$ biological parents. We can say that our grandparents are ${\displaystyle 2^{2}}$ mathematically as the number of our ancestors doubles with each generation we go back.
So:
3.a How many times would 2 be multiplied to determine the number of great grandparents?
3.b How many times would 2 be multiplied to determine the number of great-great grandparents?
3.c How many people would be our 28 ancestors?

3.a If our grandparents are the 22 generation, then our great-grandparents are one more back so are 23.

3.b This means our great-great grandparents are one more generation back so would be our 24 ancestors.

3.c Our 28 ancestors would be ${\displaystyle 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2=256}$ people.


## Problem 4

4.a Identify the square numbers between 50 and 100
4.b Identify the square numbers between 160 and 200.

4.a We know that 102 = 100 so any larger number is out of the range. So 92 = 81 in the range
82 = 64 in the range    72 = 49 too small for the range.


So the square numbers in the range are 64, 81 and 100.

4.b 122 = 144 and is too small for our range   132 = 169 is in the range   142 = 196 is in the range, and the next square will be too large.


So the square numbers in this range are 169 and 196

## Problem 5

You tear a piece of paper in half 5 times. How many scraps of paper are you left with?

You should have 32, or ${\displaystyle 2^{5},}$ scraps of paper left over. The general solution is:
${\displaystyle P=2^{r}}$, where P is the number of paper scraps and r is the number of times the paper has been torn.

An interesting anecdote related to this problem: it was once commonly believed that a piece of paper could only ever be folded 8 times; thus, when unraveled, the paper could contain at most 256 (${\displaystyle 2^{8}}$) sections. However, this was later proven false when Britney Gallivan folded a paper 12 times, and derived a function for the actual number of folds that could occur.