SECTION
10.1: WHAT ARE ALGEBRAIC EXPRESSIONS?
An "algebraic expression" is made up of constants, variables, coefficients times one or more variables, and the operations of addition, subtraction, multiplication and division.
ex. 1) 5x
ex. 2) 2xy  5
ex. 3) 4x²  3x + 10
ex. 4) (x  y  3z)/(4x²  7x + 12)
SECTION 10.2: EVALUATING ALGEBRAIC EXPRESSIONS
You must understand, as explained in chapter 3, that a variable is a box that holds a number. So when you see a variable I think it best that you think of a variable as a number.
THINK THE METAPHORS

A VARIABLE IS A BOX HOLDING A NUMBER
OR
A VARIABLE IS A NUMBER

You are going to learn how to "evaluate an algebraic expression." It is important for you to understand the RULES OF THE GAME used in evaluating an algebraic expressions.
Definition: "Evaluating an algebraic expression" means to replace all variables in the algebraic expression with numbers that have been assigned to the variables, then computing the results.
RULES FOR EVALUATING ALGEBRAIC EXPRESSIONS 
RULE 1) Any variable in an algebraic expression can be replaced with a number that has been "assigned to the variable." A number assigned to a variable is called the "value of the variable."
RULE 2) If a number is immediately followed by a one or variables, the operation of multiplication is understood to be between the number and the variable(s).
Examples of what rule 2 is saying:
ex. 1) 5x means "5 times x" and the variable x is a box containing a number.
ex. 2) 5xyz means "5 times x times y times z" and the variables x, y and z are boxes containing numbers.
RULE 3) Order of operations, PEMDAS, must be used when computing an algebraic expression.

EXAMPLES
ex 1) Evaluate 5x if x = 5
Comment: "5x" is the algebraic expression. Since there is no space between the 5 and the x the operation is understood to be multiplication. Hence 5x means "5 times x."
5" is the value of the variable x. Replace the x in 5x with the value of the variable 5. Since
ans:
5x means "5 times x."
x = 5
Hence, 5x means "5 times 5."
ans: 25
ex. 2 Evaluate 2xy  5 if x = 2 and y = 5
comment: "2xy  5" is the algebraic expression and "2" is the value of the variable x and "5" is the value of the variable y. Replace the x with 2 and y with 5 in 2xy  5.
ans:
2(2)(5)  5
4(5)  5
20  5
25
ROOTS WITH INDEXES THAT ARE POSITIVE EVEN INTEGERS 
If "a" is a positive even integer and "b" is a positive real
number
then means
"What positive real number raised to the power of "a" will result in
"b"?" 
ex. 1) √ 4 is asking "What positive real number raised to the power of 2 will result in 4?"
The answer is 2.
Why? Because 2^{2} = 4.
Hence, we say √ 4 = 2.
Comment: Remember if there is no index in the root, the root is assumed to have an index of 2.
ex. 2) is asking "What positive real number raised to the power of 4 will result in 81?"
The answer is 3.
Why? Because 3^{4} = 81.
Hence, we say = 3.
ex. 3) is asking "What positive real number raised to the power of 6 will result in 64?"
The answer is 2.
Why? Because 2^{6} = 64.
Hence, we say = 2.

ROOTS WITH INDEXES THAT ARE POSITIVE ODD INTEGERS 
If "a" is a positive odd integer and "b" is any real number
then means
"What real number raised to the power of "a" will result in
"b"?" 
ex. 1) is asking "What real number raised to the power of 3 will result in 8?"
The answer is 2.
Why? Because 2^{3} = 8.
Hence, we say = 2.
ex. 2) is asking "What real number raised to the power of 5 will result in 32?"
The answer is 2.
Why? Because (2)^{5} = 32.
Hence, we say = 2.
Comment: Roots with odd indexes can have radicands that are negative and answers that are negative. 
SECTION 10.3: REAL WORLD ALGEBRAIC EXPRESSIONS
Sir Isaac Newton 
Sir Isaac Newton, a genius in mathematics and physics, created the now famous algebraic expression below to model the force of attraction between two "bodies."
NEWTON'S LAW OF GRAVITY
(Gm₁m₂) / r²
m_{1 } = Mass of one body (kg)
m_{2 } = Mass of the second body (kg)
G = 6.672×10^{11 } (Nm²) / (kg²) (Universal Gravitational Constant)
r = Distance between the centers of mass of the two objects
FORCE BETWEEN THE EARTH AND THE SUN
m_{1 }= mass of the sun = 1.98892 × 10^{30 } kilograms
m_{2 } = mass of Earth = 5.9742 × 10^{24 } kilograms
Approximate distance between the earth and the sun = 150,000,000 km
Force between Earth and the Sun
(6.672×10^{11 } Nm² / kg²)(1.98892 × 10^{30}kg)_{ }(5.9742 × 10^{24} kg)_{ }) / 150,000,000 km²
Celcius and Fahrenheit are two types of measuring systems to measure temperature. In most of the world tempterature is measured in degrees Celcius, though in the US tempterature is measured in degrees Fahrenheit. kWater freezes at 32 degrees Fahrenheit but 0 degrees Celcius.
Many algebraic expressions produce results that tell you something about the world and our experiences. The formula (5(f  32)) / 9 will produce a Celsius temperature if you are given a value for "f," the Fahrenheit temperature.
So if it's 77 degrees Fahrenheit outside and you would like to know the Celsius temperature all you have to do is substitute 77 in the above formula for the variable "f," which represents the Fahrenheit temperature .
(5(7732))/9= 25 degrees Celsius
In Accelerated PreCalculus and Physics you will deal with many real world algebraic expressions that will need to be evaluated.



THE SUMMER STUDY GUIDE
BY CHAPTERS

RETURN TO THE SUMMER STUDY GUIDE MAIN PAGE
 CHAPTER 1: THE NUMBER SYSTEM
 CHAPTER 2: ORDER OF OPERATIONS
 CHAPTER 3: VARIABLES, MONOMIALS,
BINOMIALS, TRINOMIALS, POLYNOMIALS,
COEFFICIENTS, TERMS AND LIKE TERMS
 CHAPTER 4: SIGNED NUMBERS,
ABSOLUTE VALUE, AND INEQUALITY SYMBOLS
 CHAPTER 5: FACTORS, COMMON
FACTORS, LEAST COMMON FACTORS AND GREATEST COMMON FACTORS
 CHAPTER 6: PROPERTIES OF NUMBERS
 CHAPTER 7: THE WORLD OF FRACTIONS
 CHAPTER 8: EXPONENTS
 CHAPTER 9: ROOTS
 CHAPTER 10: ALGEBRAIC EXPRESSIONS
 CHAPTER 11: CARTESIAN COORDINATE SYSTEM
 CHAPTER 12: SETS, RELATIONS AND FUNCTIONS
 CHAPTER 13: AVERAGE RATE OF CHANGE OF Y WITH RESPECT TO X, SLOPE, PYTHAGOREAN THEOREM, AND DISTANCE FORMULA BETWEEN TWO POINTS
 CHAPTER 14: XINTERCEPT(ZERO) AND Y INTERCEPT(B)
 CHAPTER 15: LINES
 CHAPTER 16: FUNCTIONS
 CHAPTER 17: MULTIPLYING POLYNOMIALS
 CHAPTER 18: FACTORING
 CHAPTER 19: RATIONAL EXPRESSIONS
 CHAPTER 20: SOLVING EQUATIONS
 CHAPTER 21:SOLVING INEQUALITIES
 CHAPTER 22: SOLVING A SYSTEM OF EQUATIONS
 CHAPTER 23: QUADRATICS
 CHAPTER 24: CIRCLES
 CHAPTER 25: AREAS AND PERIMETERS OF PLANE FIGURES
 CHAPTER 26: VOLUMES
