chapter10.htm

An "algebraic expression" is made up of constants, variables, coefficients times one or more variables, and the operations of addition, subtraction, multiplication and division.

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.

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

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.

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) 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 = 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 2⁶ = 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 $\root{a} \of{b}$ 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 2³ = 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.

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 .

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

ACCELERATED PRE-CALCULUS SUMMER STUDY GUIDE

CHAPTER 10: ALGEBRAIC EXPRESSIONS