An Introduction to Roots
Copyright Scott Storla 2015
Procedure – Order of Operations
Begin with the innermost grouping idea and work out;
Explicit grouping ( ), [ ], { }
Implicit grouping
Operations in the numerator or denominators of fractions.
Operations in radicands, exponents or logarithmic arguments.
1. Start to the left and work right simplifying each operation beyond the basic
four as you come to them.
2. Start again to the left and work right simplifying each multiplication or
division as you come to them.
3. Simplify all terms.
4. Start again to the left and work right simplifying each addition or
subtraction as you come to them.
Copyright Scott Storla 2015
Operations and Operators
Operation
Operator(s)
Logarithm
log ln
Exponential
e
10
Root
2
Power
Absolute value
Multiplication
 
 

Division
Addition
+
Subtraction

Copyright Scott Storla 2015
Find the base
Base
exponent
For this exponent
index
 Power
Radical symbol
Find a base
Radicand
That returns this power
2
9 3
29
3
Copyright Scott Storla 2015
Find the base
Base
exponent
2
3
2
9
 Power
9
3
Copyright Scott Storla 2015
0 0
11
4 2
9 3
16  4
25  5
36  6
49  7
64  8
81  9
100  10
3
0 0
3
11
3
8 2
3
27  3
3
64  4
3
125  5
3
1  1
3
8  2
3
64  4
3
125  5
Copyright Scott Storla 2015
3
27  3
Copyright Scott Storla 2015
A couple issues with square root
Why is
4 equal to 2?
Because 22  4
But  2   4
2
There are actually two square roots of 4.
We are interested in the positive or principal
square root.
Copyright Scott Storla 2015
How much is
4 ?
2 doesn't work since 22  4
2 doesn't work since  2   4
2
4 is not a real number.
Copyright Scott Storla 2015
Fill in the blanks using the words term, factor, sum,
product, difference, quotient, base, exponent,
power, index, radicand and root.
Given
4 15
3 64
3 is an _____, 64 is a _____, 3 64 is a
_____, 4 is a_____, 15 is a_____ and a_____, 4 15 is
a_____ and the entire expression is a_____.
Copyright Scott Storla 2015
Fill in the blanks using the words term, factor, sum,
product, difference, quotient, base, exponent,
power, index, radicand and root.
Given

9  36

2
36 is a _____ and a _____, 9 is a
_____, 2 is an _____, 9  36 is a _____,

_____, and the entire expression is a _____.
Copyright Scott Storla 2015

9  36 is a
Fill in the blanks using the words term, factor, sum,
product, difference, quotient, base, exponent,
power, index, radicand and root.
Given
16 81
16 81 is both a_____ and a_____,
16 is both a_____ and a_____, 81 is a_____, and the
entire expression is a _____.
Copyright Scott Storla 2015
Fill in the blanks using the words term, factor, sum,
product, difference, quotient, base, exponent,
power, index, radicand and root.
Given
12  10 2   9  122
12  10 is a _____, (12  10) is
a _____, (12  10)2 is both a _____, and a _____,
12  10 2   9  12 2 is both a _____, and a _____, 2 is an
_____, and the entire expression is a _____.
Copyright Scott Storla 2015
Fill in the blanks using the words term, factor, sum,
product, difference, quotient, base, exponent,
power, index, radicand and root.
Given
3  ( 3)2  4  1 5 
2  1
4 is a____, ( 3) is a_____,
( 3)2 is both a_____ and a_____, ( 3)2  4  1 5  is both
a_____ and a_____, 2  1 is a _____, 3 is a_____ ,
( 3)2  4  1 5  is both a_____ and a_____,
3  ( 3)2  4  1 5  is a_____ , 4  1 5  is a_____ and
the entire expression is a_____.
Copyright Scott Storla 2015
Count the number of operators, discuss the
order and then simplify the expression.

4 9

4 9
 2  3  2  3 
5  1
5
Copyright Scott Storla 2015

Count the number of operators, discuss the
order and then simplify the expression.

9  36
3  6

2
2
92
81
Copyright Scott Storla 2015
Count the number of operators, discuss the
order and then simplify the expression.
4 36 9  3 8
4(6)(3)  2
24(3)  2
72  2
70
Copyright Scott Storla 2015
Count the number of operators, discuss the
order and then simplify the expression.
62  12  3  2
3
27
36  12  3  2(3)
36  4  6
32  6
26
Copyright Scott Storla 2015
Procedure – Order of Operations
Begin with the innermost grouping idea and work out;
Explicit grouping ( ), [ ], { }
Implicit grouping
Operations in the numerator or denominators of fractions.
Operations in radicands, exponents or logarithmic arguments.
1. Start to the left and work right simplifying each operation beyond the basic
four as you come to them.
2. Start again to the left and work right simplifying each multiplication or division
as you come to them.
3. Start again to the left and work right simplifying each addition or subtraction
as you come to them.
Copyright Scott Storla 2015
Count the number of operators, discuss the
order and then simplify the expression.
32  4  2  2 
9  4  2  2 
9   16 
9  16
25
5
Copyright Scott Storla 2015
Count the number of operators, discuss the
order and then simplify the expression.
 3  5    3  9 
2
 8 2   6 2
64  36
100
10
Copyright Scott Storla 2015
2
Count the number of operators, discuss the
order and then simplify the expression.
3
64
3

25  9
64  5  3 
3
64  2 
4  2

2
2
2
44
16
Copyright Scott Storla 2015
2
Count the number of operators, discuss the
order and then simplify the expression.
3
52  32
2
3
25  9
2
16
2
3
3
8
2
Copyright Scott Storla 2015
Count the number of operators, discuss the
order and then simplify the expression.
 7 2  4  2  5 
49   40 
9
3
Copyright Scott Storla 2015
Count the number of operators, discuss the
order and then simplify the expression.
6  62  4  4  2 
62
2  4
6  36  4  4  2
8
8
2  4
2  4
6  36  32
2  4
6 4
2  4
Copyright Scott Storla 2015
1
Count the number of operators, discuss the
order and then simplify the expression.
9  102  4  3  7 
9  4
2  3 
9  100  4  3  7 
13
6
2  3 
2  3 
9  100  84
2  3 
9  16
2  3 
Copyright Scott Storla 2015
13
6
Count the number of operators, discuss the
order and then simplify the expression.
5
7 3
5
3
4
9 1
 33 1
52  32
25  9
16
4
Copyright Scott Storla 2015
The Order of Operations
Roots
Copyright Scott Storla 2015