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ACCELERATED PRE-CALCULUS SUMMER STUDY GUIDE

CHAPTER 10: ALGEBRAIC EXPRESSIONS


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 $\root{a} \of{b}$ 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 22 = 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) cube8 is asking "What positive real number raised to the power of 4 will result in 81?"
The answer is 3.
Why? Because 34 = 81.
Hence, we say cube8= 3.

ex. 3) cube8 is asking "What positive real number raised to the power of 6 will result in 64?"
The answer is 2.
Why? Because 26 = 64.
Hence, we say cube8= 2.

 

ROOTS WITH INDEXES THAT ARE POSITIVE ODD INTEGERS

If "a" is a positive odd integer and "b" is any real number

then $\root{a} \of{b}$ means

"What real number raised to the power of "a" will result in "b"?"


ex. 1) cube8 is asking "What real number raised to the power of 3 will result in 8?"
The answer is 2.
Why? Because 23 = 8.
Hence, we say cube8= 2.

ex. 2) cube8 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 cube8= -2.

Comment: Roots with odd indexes can have radicands that are negative and answers that are negative.

 

 

SECTION 10.3: REAL WORLD ALGEBRAIC EXPRESSIONS

 

newton
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

m1 = Mass of one body (kg)

m2 = 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

m1 = mass of the sun = 1.98892 × 1030 kilograms

m2 = mass of Earth = 5.9742 × 1024 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 × 1030kg) (5.9742 × 1024 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(77-32))/9= 25 degrees Celsius

In Accelerated Pre-Calculus and Physics you will deal with many real world algebraic expressions that will need to be evaluated.

 

-----------------------------
THE SUMMER STUDY GUIDE
BY CHAPTERS

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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: X-INTERCEPT(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