Evaluate Exponents:
Natural Number Exponent: If a is a real number and n is the natural number, then
a^{n} = a * a * a * … n times (a is multiplied n times)
a is called the base and n is called the exponent.
2 * 2 * 2 * 2 is expanded form
2^{4} is in exponential form
Note: Exponents are sometimes referred as powers.
Zero as an Exponent:
Zero as an exponent a0 = 1 for all a ≠ 0
Example:
3^{0} = 1
(-41)^{0} = 1
(23/12)^{0} = 1
0^{0 }does not exist or is said to be undefined
Practice Problems:
Write in exponential form:
9 * 9 * 9 * 9 * 9 * 9 * 9 = 9^{7}
(-3) * (-3) * (-3) * (-3) * (-3) = (-3)^{5}
(4/7) * (4/7) * (4/7) * (4/7) = (4/7)^{4}
Write in expanded form:
5^{3} = 5 * 5 * 5
(-7)^{8} = (-7) * (-7) * (-7) * (-7) * (-7) * (-7) * (-7) * (-7)
Calculate value using calculator:
3^{7} = 2187
(⅙)^{4} = (1/1296)
(1.42)^{3} = 2.863288
Scientific Notation:
Scientific Notation: A number in the form a * 10^{n} where 1 ≤ |a| < 10 and n is an integer is said to be in scientific notation.
When a number is so large that it cannot be displayed all at once on the calculator screen; th calculator displays it in scientific notation.
Try 8^{13} = 5.437558139E11 or 5.437558139^{11} (exponent on 10)
This is same as 5.437558139 * 10^{11} = 543755813900
543755813900 is the standard form
Convert from Scientific Notation to Standard Form:
The exponent on 10 is positive n: Multiply by 10^{n}, which is equivalent to moving the decimal point n places to the right.
Example:
4.53 * 10^{5} = 4.53 * 100000 = 453000
-2.748 * 10^{4} = -2.748 * 10000 = 27480
Order of Operations:
PEMDAS (L to R)
Evaluate everything inside the grouping symbols first, inside to outside working from left to right
Evaluate all exponents and divisions working from left to right
Do all multiplications and divisions in the order they occur, working from left to right
Do all additions and subtractions in the order they occur, working from left to right
Examples:
8/4 - 16 + 9 * 3 - 52
= 2 - 16 + 27 - 25
= -12
Commutative Property:
If a and b are real numbers, then the commutative properties are as follow:
Commutative property of addition is a + b = b + a
eg: 8+5 = 5+8
Commutative property of multiplication is a * b = b * a
eg: 3*5 = 5*3
Commutative word comes from french word means “to switch or substitute”
Associative Property:
If a, b & c are any real numbers, then the associative properties are as follows:
Associative property of addition is a + (b + c) = (a +b) + c
eg: 2 + ( 5 + 8) = (2 + 5 ) + 8
Associative property of multiplication is a * (b * c) = (a * b) * c
eg: -2 * ( 3 * 8) = (-2 * 3) * 8
Distributive Property:
If a, b, c are real numbers, then the distributive property states:
a(b + c) = a * b + a * c
a(b - c) = a * b - a * c
Example:
5 * (6 + 7) = 5 * 6 + 5 * 7
5 * ( 6 - 7 ) = 5* 6 - 5 * 7
Constant, Variable, Coefficient:
Variable: A variable is a symbol, usually a letrer, used to represent a quantity
Note: O is not a recommended letter to represent variable as it can be easily confused with number 0
Constant: A constant is a value that does not change
Coefficient: The coefficient of a variable term is the numercial factor of the term
Example: 4x has 4 as coefficient
Problem:
Stephanie earns $9.25 per hour working as a tutor. Her weekly pay can be computed as 9.25h, where h is the number of hours she worked in a week.
If Stephanie works 12 hours in a week, what is her weekly pay?
If stephanie works 26 hours in a week, what is her weekly pay?
Evaluating Expressions:
Substitute the given value of variable into the expression
Simplify using the order of operations agreement (PEMDAS - L to R)
Example:
12r - 16 for r = 1/2
8(n + 3) - 15/ (n + 2) for n = 3
Unit Conversion:
Unity Fraction: A unity fraction is a fraction that has units in the numerator and deniminator and whose simplified values is equal to 1. Also known as conversion factors.
Example:
100cms = 1 m,
100cm/1m is a unity fraction
Find the unity fraction.
Steps to convert units:
Always place unit that are wanted in final answer in the numerator
Place the other unit (to be converted) in the denominator
Multiply the given value by unity fraction. Divide out the common units
Reduce to the lowest terms
Examples:
Convert 254 feet to meters (1m ≈ 3.28 feet)
254 ft * 1m/3.28ft = 77.44m
Convert 15 gallons to liters (1 gal = 3.785 l)
15 gal*3.785 l/1 gal = 56.775 l
Defining Variables:
Write a sentence that explains what the variable represents
Weekly pay check for an hourly paid employee
a = amount of pay in dollars
h = Number of hours worked in a week/month
Time you drive from San Diego to Los Angeles depends on the speed
t = time to drive from SD to LA in hours
s = speed of the car in mph
d = distance (miles)
Practice Problems:
The amount of money (in dollars) a family spends a week on gas depends on the number of miles they drive that week.
a = amount of money in dollars
n = number of miles driven
The temperature outdoors (in degrees Fahrenheit) depends on the time of the day
F = temperature in Fahrenheit
t = time of the day
Translating Phrases into Algebraic Expressions:
Quotient of a number and 12 → n/12
The product of 7 and sum of a number and 12 → 7(n+12)
The sum of 7 divided by a number and 2 → 7/x + 3
The difference between -3 times a number and 8 → -3x-8
Five times a number divided by 12 → 5x/12
Generating Expressions from Tables:
Many times a input-output table is used to generate an expression
Input is independent variable - data to be entered in the table. The independent variable is not changed or controlled by another variable.
Output is dependent variable - results based on input. The dependent variable tends to change by the independent variable.
Like Terms:
Term: A constant (a number) or a product of a constant and a variable (or variables) is called a term.
Example: -4a, ⅔ km2, x, 3
Like Terms: Two or more terms that have the same variable part, where the variables are raised to the same exponents, are called like terms.
Example: 3x and 5x
-25a2bc3 and 10a2bc3
Practice Problems:
Identify the number of terms and the like terms in each expression
2x + 8 + 7x
3 terms; like terms: 2x & 7x
7xy + 8x - 2xy - 3y
4 terms; like terms: 7xy & 2xy
t^{5} + 4t^{3} + 3t^{5} - t^{3}
4 terms; like terms: t^{5} & 3t^{5} , 4t^{3} & t^{3}^{}^{}
Addition & Subtraction Properties:
Two or more terms can be added or subtracted only if they are like terms. Add or subtract the coefficients of the like terms. The variable part will remin the same.
5x +7x = 12x
-25a^{2}bc^{3} + 10a^{2}bc^{3} = -15a^{2}bc^{3}
Practice Problems:
7y + 20y = 27y
a - 4b - 6a + 5b = -5a + b
-3b + 11 - 7b -9 = -10b + 2
5y^{2} - 3y - 2 + 8y^{2} - 7y - 9 = 13y^{2} - 10y - 11
^{}
Multiplication & Distributive Properties:
When multiplying a single term by a constant, multiply the coefficient by the constant, and the variable parts stay the same
5(4xy) = 20xy
When multiplying a multiple term expression by a constant, multiply the coefficient of each term by the constant, and the variable parts will stay the same
3(2x-7y) = 6x - 21y
Practice Problems:
6(9w^{3}) = 54w^{3}
7(3g^{5} + 5g^{3} + 2g - 8) = 21g^{5} + 35g^{3} + 14g - 56
-3(12st + 5t^{2} - 9s) = -36st -15t^{2} + 27s
Simplifying Expressions:
Simplify following expressions
4(2c + 5) - 3c + 11 = 5c + 31
5m + 7(3n - 4) - (6m + 2n -18) = -m + 19n -10
⅓(6b - 15a) - ¼(16a - 8b) = -9a + 4b
2.1(3b + 5) + 6.1b +10 = 12.4b + 20.5
Reading a Table:
The table gives the data on the number of days on which ozone levels exceeded the EPA’s health based standard
What is the maximum number of days on which the ozone levels exceeded the EPA’s health based standard? When did this occur?
In what year were there 5 days on which ozone levels exceeded the EPA’s health based standard?
Bar Graphs:
Data are easier to examine if they are visually presented in a graph.
Horizontal axis is called x-axis. This is where input values are plotted.
Vertical axis is called y-axis. This is where output values are plotted.
Example of a bar graph:
Height of each bar
represents the output value
for corresponding input vlaue
Reading Bar Graphs:
For which month did 6 students have their birthday?
For which months less than 2 students have their birthdays?
How many students had their birthdays in December?
Creating Bar Graph:
- The input values will be plotted on x-axis and output values on y-axis
- Evenly and consistently space the x-axis so that it includes the input values (data)
- Evenly and consistently space the y-axis so that it includes the output values (data)
- Draw the vertical bars of uniform width. Center each bar over the appropriate input value. Draw the bars of height corresponding to the output value.