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MAT 101 – Lecture 2 Notes
Definitions from the text – Sections 1.5
to 2.1
1.5 – Exponential Expressions and the Order of Operations

Arithmetic expressions are meaningful combinations
of numbers


Ex: 3·2+6
Sometimes, grouping symbols like parentheses,
brackets, absolute value symbols, and fraction bars
are used to indicate which operation must be
performed first in an expression


Ex: (3+2)·5 or 3+(2·5) mean different things!
Ex:
is treated as if the numerator and denominator
35
are each
8 in
2 parentheses
1.5 – Exponential Expressions and the Order of Operations
Example 1: Evaluate each expression.

a)
b)
c)
(3-8)(4-5)
|3-4|-|5-(-2)|
4 - (-2)
5-8
Exponential expressions are a type of arithmetic
expression of the form an, as defined below:


For any counting number n, and real number a
a n  a 
a  a 
...  a
n  factors
where a is the base and n is the exponent.

The expression an is read “a to the nth power” and when
n=1, the power is often omitted.
1.5 – Exponential Expressions and the Order of Operations

Example 2: Write each product as an
exponential expression.
a)
b)
c)

6·6·6·6·6
(-3)(-3)(-3)
2/3 ·2/3 ·2/3 ·2/3
Examples 3-4: Write each exponential
expression as a product without exponents,
then evaluate.
a)
34
b) (-2)2
c) (5/4)2
d) (-0.1)3
1.5 – Exponential Expressions and the Order of Operations

What is the difference between –24 and
2)4?



(-
An exponential expression with a negative base
has parentheses written around the base, while
the ‘opposite’ of an exponential expression with a
positive base has the negative written in front of
the base.
(-2)4 = (-1)4(2)4 = 16.
-24 = (-1)(24) = -16
1.5 – Exponential Expressions and the Order of Operations


Example 5: Evaluate.
a)
(-10)4
b)
-104
c)
-(-0.5)2
d)
-(5-8)2
The order of operations:
(Good Sir, Excuse My/Dear Aunt/Sally)
1) Evaluate expressions within all GROUPING SYMBOLS first
2) Evaluate EXPONENTIAL expressions (any order)
3) Perform MULTIPLICATION and DIVISION (any order)
4) Perform ADDITION and SUBTRACTION (any order)
1.5 – Exponential Expressions and the Order of Operations
Example 6: Evaluate using the order of
operations.

a)
c)
23 · 32 b) 2 · 5 – 3 · 4 + 42
2 · 3 · 4 – 33 + 8
2
Example 7 (grouping symbols)

a)
3 – 2(7 – 23)
b) 3 - |7 - 3 ·4|
c)
–9 –58
 52  3(7)
Example 8 (grouping within groupings)

a)
6 – 4[5 – (7 – 9)]
b) -2|3 – (9 – 5)| - |-3|
1.5 – Exponential Expressions and the Order of Operations

Example 9

A strategy among gamblers is to “double down”. A gambler
loses $100 and then begins to employ this strategy. He
then loses 4 more times in a row.






Model this situation with an exponential expression.
What is his next bet amount going to be?
How much money has he lost before making this next bet?
$100 · 2 · 2 · 2 · 2 = $100 · 24
$100 · 25 = $3,200
$100 + $100 · 2 + $100 · 22 + $100 · 23 + $100 · 24 = $100 +
$200 + $400 + $800 + $1,600 = $3,100
1.6 – Algebraic Expressions



Variables (symbolized by letters) are used to represent
numbers.
An arithmetic expression contains a combination of numbers
and variables with the operations of arithmetic.
Expressions are often named after the last operation that is to
be performed in the expression.





x + 2 [sum]
a – bc [difference]
3(x – 4) [product]
(a + b)2 [square]
[quotient]
3
x4
1.6 – Algebraic Expressions
Example 1: Identify each expression as either a
sum, difference, product, quotient, or square.

a)
b)
c)
d)
3(x + 2)
b2 – 4ac
(a – b)2
•
Product
Difference
Square
Quotient
•
•
•
ab
cd
1.6 – Algebraic Expressions

Sample Verbal Expressions and their Corresponding
Algebraic Expressions:





The sum of 5x and 3  5x + 3
The product of 5 and x + 3  5(x + 3)
x
x
The sum of 8 and
 8+
3
3
The difference of 3 and x2  3 – x2
The square of 3 – x  (3 – x)2
1.6 – Algebraic Expressions
Example 2: Translate each algebraic expression
into a verbal expression using the word sum,
difference, product, quotient, and/or square.

a) 3/x
d) (a + b)2
•
•
•
•
•
b) 2y + 1
e) (a – b)(a + b)
c) 3x – 2
The quotient of 3 and x
The sum of 2y and 1
The difference of 3x and 2
The square of the sum of a + b
The product of a – b and a + b
1.6 – Algebraic Expressions
Example 3: Translate each verbal expression into
an algebraic expression.

a)
b)
c)
d)
The quotient of a + b and 5
The difference of x2 and y2
The product of π and r2
The square of the difference x – y




ab
5
2
x – y2
πr2
(x – y)2
1.6 – Algebraic Expressions


When a variable is used in an algebraic
expression, the value of the expression
hinges on the value given to the variable.
If we are given the value of a variable, we
can evaluate an expression with that variable
in it by replacing the variable with the value
given.


Ex: For a = -2, 2a = 2(-2) = -4;
Ex: For a = 2, a – 5 = 2 – 5 = -3
1.6 – Algebraic Expressions
Example 4; Evaluate each expression using a = 3, b = 2, c = -4.

a) 2a + b – c
c) b2 – 4ac
b) (a – b)(a + b)
d)
 a 2  b2
c b
So far, we have focused our attentions on
expressions. An equation is a statement of
equality between two expressions.


Ex: 11 – 5 is an expression, and 11 – 5 = 6 is an
equality.
1.6 – Algebraic Expressions


In an equation that includes a variable, any number
that gives a true (equal) statement when substituted
for the variable is said to satisfy the equation.
That number is also referred to as a solution (or
root) to the equation.


Ex: x + 3 = 9. What is the solution to this equation?
We can translate equalities to verbal expressions by
using “is” or “is the same as” or “is equal to” instead
of the equality symbol (=).
1.6 – Algebraic Expressions
Example 5: Determine whether the given number
is a solution to the equation by substituting the
number for x.

a)
6, 3x – 7 = 9 b)
2x  4
-3, 5  2
c) -5, -x – 2 = 3(x + 6)
Example 6: Translate each sentence into an
equation.

a)
b)
c)
The sum of x and 7 is 12.
The product of 4 and x is the same as the sum of y and
5.
The quotient of x + 3 and 5 is equal to -1.
1.6 – Algebraic Expressions
Example 7: Using algebraic expressions to
model real-life situations.

A forensic scientist uses the expression 69.1 +
2.2F as an estimate of the height in centimeters
of a male with a femur of length F centimeters
(National Space Biomedical Research Institute,
www.nsbri.org)

a)
If the femur of a male skeleton measures 50.6 cm,
then what is the person’s height?
1.6 – Algebraic Expressions
b) Using the graph below, estimate the length of a
femur for a person who is 150 cm tall.
200
190
Height of Person (cm)
180
170
160
150
140
130
120
110
100
20
30
40
Length of Femur (cm)
50
60
1.7 – Properties of the Real Numbers


What is the price of a hamburger for you and fries,
and a Coke for your friend? Does the order in which
the cashier rings these in matter? Does the
grouping (for you or your friend) matter? Why?
Commutative properties of addition and
multiplication (for real numbers a and b):



Associative properties of addition and multiplication
(for real numbers a,b,c):



a+b=b+a
ab = ba
(a + b) + c = a + (b + c)
(ab)c = a(bc)
Do these properties hold for subtraction or division?
1.7 – Properties of the Real Numbers

Examples 1-2: Rewrite each expression
using the commutative properties (of
addition and multiplication).
a)
b)
c)
d)
e)
f)
2 + (-3)
n·3
5 – yx
2y – 4x
(x + 2) · 3
8 + x2
1.7 – Properties of the Real Numbers

Example 3: Use the commutative and
associative properties of multiplication and
exponential notation to rewrite each
expression.
a)
b)

(-3y)(y)
xy(5yx)
While subtraction does not have
commutative or associative properties,
every subtraction can be rewritten as
addition of an additive inverse…
1.7 – Properties of the Real Numbers
Example 4: Rewrite subtractions as additions of
the additive inverses and apply the properties of
addition to solve.

a)
b)

3–7+9–5
4 – 5 – c + 6 – 2 + 4c – 8
The distributive property for real numbers a,b,c is
stated as follows:
a(b + c) = ab + ac


With the aid of this property, we can remove
parentheses and rewrite expressions as sums or
differences.
Similarly, we can use this property to factor out a
common multiple.
1.7 – Properties of the Real Numbers

Example 5: Use the distributive property to
remove parentheses.
a)
b)

a(3 – b)
-3(x – 2)
Example 6: Use the distributive property to
factor each expression (insert parentheses).
a)
b)
7x – 21
5a – 5
1.7 – Properties of the Real Numbers



The additive identity is 0 because addition of
0 to any number does not change that
number.
The multiplicative identity is 1 because
multiplication of 1 and any number does not
change that number.
Identity properties (of addition and
multiplication) for real number a:


a+0=0+a=a
a·1=1·a=a
1.7 – Properties of the Real Numbers



Recall that every real number a has an
additive inverse (or opposite), -a, such that a
+ (-a) = 0
Similarly, for any nonzero real number a,
there exists a multiplicative inverse (or
reciprocal), 1/a, such that
a · (1/a) = 1
These properties are known as the additive
inverse property and multiplicative inverse
property (respectively).
1.7 – Properties of the Real Numbers

Notice also that 0 has a special property in
multiplication. What is it?



Multiplication of any real number with zero
ALWAYS results in 0! ( 0 · a = 0)
This is called the multiplication property of
zero.
List all of the properties we have learned
today and give an example as a class.
1.7 – Properties of the Real Numbers
Example 7: Find the multiplicative and additive
inverse of each number.

a)
5
b) 0.3
c) -0.75
d) 1.91
Example 8: Name the property that explains each
equation.

a)
c)
e)
f)
g)
i)
5·7=7·5
b) 4 · (¼) = 1
1 · 864 = 864
d) 6 + (5 + x) = (6 + 5) + x
3x + 5x = (3 + 5)x
6 + (5 + x) = 6 + (x + 5)
325 + 0 = 325
h) -5 + 5 = 0
455 · 0 = 0
j) 4(x2 + y2) = 4x2 + 4y2
1.8 – Using the Properties (of Real Numbers) to Simplify Expressions




The properties we have learned about in section 1.7
can be useful in simplifying computations (see
Example 1).
In particular, the distributive property is useful for
allowing us to combine like terms.
Terms are the parts of an expression separated by
the operations (+, - , ×, ÷). They are comprised of a
number or the product of a number and one or more
variables raised to powers. E.g. -3, 5x, 3x2y, -abc
are terms
Like terms are those with the same variables AND
the same exponents of those variables
1.8 – Using the Properties (of Real Numbers) to Simplify Expressions
The number preceding the variables in a
term is referred to as the coefficient.

Ex: -5ab2 has a coefficient of -5
Ex: -3, 5x, 3x2y, -abc have coefficients -3, 5, 3,
and -1 (respectively)



Example 2: Use the distributive property to
perform the indicated operations.
a)
b)
3x + 5x
-5xy – (-4xy)
1.8 – Using the Properties (of Real Numbers) to Simplify Expressions

Example 3: Combine like terms where possible.
a) w + 2w
1
1
x

d) 2 4 x


b) 7xy – (-12xy)
c) 2x2 + 3x3
Simplifying an expression involves performing
operations, combining like terms, and arriving at an
equivalent expression (that generally looks
“simpler”).
“Simplify” is a term that we use loosely, as there are
often multiple ways to cite an expression – which
are all considered “simplified”.

Ex:
x
versus 1 x - Neither is considered “simpler”
2
2
1.8 – Using the Properties (of Real Numbers) to Simplify Expressions

Examples 4-8: Simplify.
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
3(5x)
x
2( 2)
b
(-3a)( )
3
(-4a)(-7ab)
10x
5
4 x  10
2
-(x – 4) + 5x – 1
10 – (x – 3)
-2x(3x – 7) – 3(x – 6)
x – 0.02(x + 500)
1.8 – Using the Properties (of Real Numbers) to Simplify Expressions

Example 9: Perimeter of a rectangle.

Find an algebraic expression for the perimeter of
the rectangle shown here and then find the
perimeter if x = 15 inches.

Recall that the perimeter is the sum of the length of the
sides of any figure.
2.1 – The Addition and Multiplication Properties of Equality


If two students have the same number of
blank pages in their notebooks, and they
each use the same number of pages to take
notes today, they will have the same number
of blank pages in their notebooks at the end
of class!
Think of an equation like a balance scale. In
order for the scale to keep balanced, the
same operations must be performed to both
sides.
2.1 – The Addition and Multiplication Properties of Equality

This “balance” concept can be summarized by the
addition and multiplication properties of equality for
real numbers a,b,c.
Addition Property of Equality:
Adding the same number to both sides of an equation does
not change the solution to the equation. So, if a = b, then
it is true that
a+c=b+c

Multiplication Property of Equality (c ≠ 0):
Multiplying both sides of an equation by the same nonzero
number does not change the solution to the equation. So,
if a = b, then it is true that
ac = bc

2.1 – The Addition and Multiplication Properties of Equality


The “balance” concept will be used
throughout the class when solving equations
and inequalities.
A linear equation in one variable x is an
equation of the form
ax = b,

where a and b are real numbers and a≠0.
All Examples (1-9) – as a class