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Algebra

A.1 (ABCD) Prove that the following expression can be written as a product between $a-d$ and $b+c$

$ab-cd+ac-bd$

A.2 (Using Properties of Numbers) Justify each step, using the properties of communativity and associativity in proving the following identities.

$a.\ (a+b)+(c+d)=(a+d)+(b+c)$
$b.\ (a+b)+(c+d)=(a+c)+(b+d)$
$c.\ (a-b)+(c-d)=(a+c)+(-b-c)$
$d.\ (a-b)+(c-d)=(a+d)-(b+c)$
$e.\ (a-b)+(c-d)=(a-d)+(c-b)$
$f.\ (a-b)+(c-d)=-(b+d)+(a+c)$
$g.\ ((a+b)+c)+d=(a+c)+(b+d)$
$h.\ (a-b)-(c-d)=(a-c)+(d-b)$

A.3 (Using Properties of Numbers) Determine if the following statements are true or false. Justify your conclusions.

a. If $a$ , $b$ , and $c$ are integers, then the number $ab+bc$ is an even number.
b. If $b$ and $c$ are odd integers, and $a$ is an integer, then the number $ab+bc$ is an even number.

A.4 (Using Properties of Numbers) We define an integer $a$ to be of

Type I if $a=4n$ for some integer $n$
Type II if $a=4n+1$ for some integer $n$
Type III if $a=4n+2$ for some integer $n$
Type IV if $a=4n+3$ for some integer $n$

a. Provide at least two examples of each of the four types of integers above.
b. Is it true that if $a$ is even, then it is of type I or III? Justify your answer.
c. Is it true that if $a*b$ is of type I, whenever $a$ or $b$ are of type III? Justify your answer.

A.5 (Huge Powers) Put the following in order from smallest to largest.

{2}^{800}

,

{3}^{500}

,

{5}^{400}

,

{6}^{300}

A.6 (Which is Bigger?) Which is bigger? ${10}^{250}$ or ${6}^{300}$ ?

A.7 (Using Properties of Numbers) For all real numbers $x$ and positive integers $n$ , show that:

(1-x)(1+x+x^{2}+...+x^{n-1}+x^{n})=1-x^{n+1}

A.8 (Magic Trick) Choose any number. Add 3 onto the number, then multiply the result by 2. Subtract the chosen number, then subtract 4, and then subtract the chosen number again. The number you end with is 2, isn't it? Why does this trick work?

A.9 (Exponentially Exciting) For each of the following, determine the first whole number x, greater than 1, for which the second expression is larger than the first.

$a.\ x^{3},\ 3^{x}$
$b.\ x^{4},\ 4^{x}$
$c.\ x^{5},\ 5^{x}$
$c.\ x^{6},\ 6^{x}$

A.10 ( $x^{n}$ vs. $n^{x}$ ) On the basis of your answers to Problem A.14, make a conjecture that appears to be true about the two expressions $x^{n}$ and $n^{x}$ , where n = 3, 4, 5, 6, 7, .... and x is a whole number greater than 1.

A.11 (Difference of Squares)

$x$	$y$	$(x+y)(x-y)$	$x^{2}-y^{2}$

1. Choose two distinct values for x and y, and then fill in the first row for the table above.

2. Compare the results you got for the two expressions. What do you think the results from part a tell you about the difference of two squares?

3. Fill in the remaining rows of the table for different values of x and y, including negative numbers. Do you think your conjecture from part (b) is correct? Explain.

A.12 (Tricky Products) Evaluate the following expressions without using a calculator:

$a.\ (101)^{2}$
$b.\ (95)^{2}$
$c.\ (998)(999)$
$d.\ (63)(57)$
$e.\ (71)^{2}$

A.13 (Inequalities) Determine what sign values on $x$ and $y$ would make the following statements true.

$a.\ xy>0$	$b.\ x^{2}y>0$	$c.\ {\frac {x}{y}}<0$
$d.\ -{\frac {x}{y}}<0$	$e.\ -{\frac {x^{2}}{y}}>0$	$f.\ {\frac {y^{3}}{x^{2}}}>0$

A.14 (Rewriting Expressions) Evaluate the following expression without using a calculator:

${\frac {2013*2014-2013*1992}{2014-1992}}$