Algebra/Arithmetic/Exponent Answers

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Problem 1

$7^{3}$
$5+4^{2}$
$1213-9^{3}$

Answers

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

Problem 2

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

Answers

 2.a  $10^{4}=10\times 10\times 10\times 10=10,000$ 

 2.b  $10^{7}=10\times 10\times 10\times 10\times 10\times 10\times 10=10,000,000$  

 2.c  $10^{1}0=10,000,000,000$

Problem 3

Everybody is born to $2^{1}$ biological parents. Our parents each had $2^{1}+2^{1}$ biological parents. We can say that our grandparents are $2^{2}$ 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?

Answer:

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  $2\times 2\times 2\times 2\times 2\times 2\times 2\times 2=256$  people.

Problem 4

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

Answer

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?

Answer

You should have 32, or $2^{5},$ scraps of paper left over. The general solution is:

 $P=2^{r}$ , 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 ( $2^{8}$ ) 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.

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