Name: _________________________________________________________ Date: _______________ Per: ______
Alg 2 / Trig – Introduction to Roots of Real Numbers
Definition of Square Root: The inverse operation to squaring a number is to find its square roots. For example,
finding a square root of 36 is equivalent to asking, “what number when squared equals 36?” One obvious answer to
this question is 6 because 6 2  36 , but -6 will also work, because  6   36 .
2
Ex) Identify the square roots of the following real numbers:
a) 25
b) -49
d) 1
e)
f)
4
25
The radical sign
c) 0
1
9
e) 121
g) .09
h) -144
is used to denote the principal square root of a number. The symbol
the principal nth root of a number. In the expression
n
3
is used to denote
a , n is called the index of the radical, and a is called the
radicand. For a square root, the index is 2, but it is usually not written is denoted simply as
index of 3 is called a cube root, denoted by
n
.
For the purpose of simplifying radicals, it is helpful to know the
following powers:
. A radical with an
Complete the statements before you begin the exercises:

4
81 is ______ because
4
 81 is ______ because
3
27 is ______ because
3
 27 is ______ because
4

 81
3  27

3  27
Ex) Simplify the expressions by identifying the principal root without a calculator, or state those that do not
represent a real number.
Just like “squaring” and taking the square root ‘undo’ each other, so do taking the nth root and raising to the nth
power. Carefully read the following definition:
Consider the example: 3  2  3  8  2 versus the example: 4  2  4 16  2 as a rationale for the need
for the absolute value of a when n is a positive even integer. Use this concept/definition to simplify the following:
3
6
4
 52
6
50 6
3
 923
a6
3
u  v3
4
 x  y 4