Definitions
Exponents and Order of
Operations
Objective: To use exponents, the
order of operations, and grouping
symbols.
variable – a symbol, usually a letter,
that represents one or more numbers
„ algebraic
l b i expression
i – a math
th phrase
h
with numbers, variables, and operation
symbols
„
Grouping Symbols
Definitions & Operation Symbols
„
„
„
„
Addition – sum,
more than, increase
Subtraction –
difference, less than,
decrease
Multiplication –
product, times, of
Division – quotient,
per
„
( ) parentheses
[ ] brackets
„{ } b
braces
„ fraction bar
a+b
„
„
„
a-b
„
a●b, (a)(b), a(b),
(a)b, ab
a÷b, a/b, a , b√a
b
„
Powers
power -- an expression like x2, where x
is the base and 2 is the exponent
„ squared
d – to
t th
the 2nd power
„ cubed – to the 3rd power
„
Try these.
Expand and multiply.
a) 34
b) 82
c) 105
d) 193
1
SOLUTIONS
Expand and multiply.
a) 34
3(3)(3)(3) = 81
b) 82
8(8)=64
8(8) 64
105
10(10)(10)(10)(10)
=100,000
d) 193
(19)(19)(19)
6859
c)
Order of Operations
1.
2.
3.
4.
Examples
P.E.M.D.A.S.
„
1.
2.
3.
4.
Please Excuse My Dear Aunt Sally!
Simplify using the Order of Operations.
e) 6 – 10 ÷ 5
f) 3 · 6 – 42 ÷ 2
„
P – parentheses
th
((any grouping
i symbols)
b l )
E – exponents
M/D – multiply/divide
A/S – add/subtract
Examples – Solutions
Simplify using the Order of Operations.
e) 6 – 10 ÷ 5
f) 3 · 6 – 42 ÷ 2
6–2
3 · 6 – 16 ÷ 2
4
18 – 8
10
„
Grouping symbols (parentheses,
brackets, fraction bar)
E
Exponents
t
Multiply and divide in order from left
to right.
Add and subtract in order from left to
right.
More Examples
Simplify using the Order of Operations.
g) 4 · 7 + 4 ÷ 22
h) 53 + 90 ÷ 10
„
2
Simplify using the Order of
Operations.
More Examples
Simplify using the Order of Operations.
g) 4 · 7 + 4 ÷ 22
h) 53 + 90 ÷ 10
4·7+4÷4
125 + 90 ÷ 10
28 + 1
125 + 9
29
134
„
Simplify using the Order of
Operations. SOLUTIONS
i) 12 + 4(2 + 3) 2
i) 12 + 4i(2 + 3) 2
Simplify using the Order of
Operations.
j) 8 − 2[(5 − 3) 2 − 1]
12 + 4(5) 2
8 − 2[(2) 2 − 1]
12 + 4(25)
12 + 100
8 − 2[4 − 1]
8 − 2[3]
k) 5 − [3i(4 − 2) 2 ] + (3i5)
l) 14 ÷ 7i[12 ÷ (4 − 2) 2 i5 − 3]
8−6
2
112
Simplify using the Order of
Operations -- SOLUTIONS
k) 5 − [3(4 − 2) 2 ] + (3i5)
5 − [3(2) ] + (15)
2
5 − [3(4)] + 15
5 − [12] + 15
− 7 + 15
8
j) 8 − 2[(5 − 3) 2 − 1]
Examples
l) 14 ÷ 7[12 ÷ (4 − 2) 2 i5 − 3]
14 ÷ 7[12 ÷ (4 − 2) i5 − 3]
2
14 ÷ 7[12 ÷ (2) 2 i5 − 3]
14 ÷ 7[12 ÷ 4i5 − 3]
14 ÷ 7[3i5 − 3]
14 ÷ 7[15 − 3]
14 ÷ 7[12]
2[12]
24
⎛3⎞
m) ⎜ ⎟
⎝4⎠
⎛ 32 ⎞
=⎜ 2⎟
⎝4 ⎠
9
=
16
2
⎛1⎞
n) ⎜ ⎟
⎝5⎠
3
⎛ 13 ⎞
=⎜ 3⎟
⎝ 5 ⎠
1
=
125
⎛1⎞
o) ⎜ ⎟
⎝2⎠
4
⎛2⎞
i⎜ ⎟
⎝3⎠
2
⎛ 14 ⎞ ⎛ 2 2 ⎞
= ⎜ 4 ⎟⎜ 2 ⎟
⎝ 2 ⎠⎝ 3 ⎠
⎛ 1 ⎞⎛ 4 ⎞
= ⎜ ⎟⎜ ⎟
⎝ 16 ⎠ ⎝ 9 ⎠
4
1
=
=
144 36
3
Examples
5
⎛ 3 ⎞⎛ 2 ⎞
− ⎜
p)
⎟⎜
⎟
9
⎝ 4 ⎠⎝ 3 ⎠
5
⎛ 6 ⎞
=
− ⎜
⎟
9
⎝ 12 ⎠
5
1
−
9
2
10
9
=
−
18
18
1
=
18
=
3 ⎛ 2 3 ⎞
i ⎟
⎜
4 ⎝ 3 5 ⎠
3 ⎛ 6 ⎞
=
⎜
⎟
4 ⎝ 15 ⎠
Examples
q)
3 ⎛ 2 ⎞
⎜
⎟
4 ⎝ 5 ⎠
6
=
20
3
=
10
=
2
4
⎛5⎞
r) ⎜ ⎟ ÷
3
⎝6⎠
25 3
=
i
36 4
75
=
144
25
=
48
2(7 + 8) + 2
3·5 + 1
2(15) + 2
=
15 + 1
32
=
16
s)
=2
4