Download Finding the square root of a number and squaring a number are

Rational Exponents
Remediation Notes
Roots and Radical Expressions will extend your knowledge of roots to include cube
roots, fourth roots, fifth roots and so on.
Simplifying Square Roots
Since 42 = 16, 4 is the square root of 16. Finding the square root of a number and
squaring a number are inverse operations. To find the square root of a number n,
you must find a number whose square is n. For example, a square root of 49 is 7
since 72 = 49. Likewise, (–7)2 = 49, so –7 is also a square root of 49.
The symbol
is called a radical sign. An expression written with a radical sign
is called a radical expression. The expression written under the radical sign is
called the radicand.
For any real numbers a and b, if a2 = b, then a is a square root of b.
NOTE: Every positive real number has two real number square roots. The number
0 has just one square root, 0 itself. Negative numbers do not have real number
square roots. When evaluating we choose the positive value of a called the
principal root.
a b
This symbol represents the principal square root of a.
The principal square root of a non-negative number is its nonnegative square root.
25  5
 a  b
To name the negative square root of a, we use this symbol.
 25  5
 a  b
To indicate both square roots, use the plus/minus sign which indicates positive or
negative.
 25  5
 a  No Real Solution
Negative numbers do not have real number square roots.
 25  No Real Solution
Not all numbers are perfect squares. However, you can simplify radial expressions
in which the radicand is not a perfect square using the Product Property of Square
Roots. Recall:
ab  a  b .
Examples:
:
A. Simplify
Steps
Explanation
50  5  5  2
 25  2
5 2
50 .
Prime Factorizat ion
(5)(5)  25 - A Perfect Square
Simplify
B. Simplify
25
147 .
Nth Roots
When there is no index number, n, it is understood to be a 2 or square root. For
example:
= principal square root of x. Not every radical is a square root.
If there is an index number n other than the number 2, then you have a root other
than a square root.
Since 32 = 9. we call 3 the square root of 9.
Since 33 =27 we call 3 the cube root of 27.
Since 34 = 81, we call 3 the fourth root of 81.
This leads us to the definition of the nth root of a number. If an = b then a is the
nth root b notated as, a  n b .
Since (-)(-) = + and (+)(+) = + , then all positive real numbers have two square roots.
Remember in our Real Number System the b is not defined. However we can
find the cube root of negative numbers since
(-)(-)(-) = a negative
one root.
and (+)(+)(+) = a positive. Therefore, cube roots only have
Below is a summary of the number of possible real roots of a number.
Type of Number
+
0
-
Number of Real nth Roots
when n is even.
2
1
None
Number of Real nth Roots
when n is odd.
1
1
1
Examples:
3
9.
Final all real cube roots of -125, 64, 0 and 9.
Solutions: -5, 4, 0 and
Find all real fourth roots of 16, 625, -1 and 0.
Solutions: ±2, ±5, Undefined, 0
As previously stated when a number has two real roots, the positive root is called
the principal root and the radical indicates the principal root. Therefore when
asked to find the nth root of a number we always choose the principal root.
Nth roots of Variables
Lets use a table to see the pattern when simplifying nth roots of variables.
x2
x x
= x
x4
x2  x2 = x • x
3
x x x
= x2
x3  x3 = x • x
=x2
3
x3
3
x6
n
xm
3
=x
=xm/n
(Note: In the first row above, the absolute value of x yields the principal
root in the event that x is negative.)
Simplifying Radical
Expressions
The general rule for reducing the radicand is to remove any perfect powers.
We are only considering square roots here, so what we are looking for is any
factor that is a perfect square. In the following examples we will assume
that x is positive.
Example:
Example:
Although x3 is not a perfect square, it has a factor of x2, which is the square of x.
Example:
Here the perfect square factor is x4, which is the square of x2.
Example:
In this example we could take out a 4 and a factor of x2, leaving behind a 2
and one factor of x.
Example:
3
80x 7 y 3
3
80x 7 y 3  3 2  2  2  2  5  xxxxxxx  yyy
 3 (2  2  2)  2  5  (xxx )(xxx )x  (yyy )
All the sets of "3" have been grouped. they are cubes!
 2x 2y
3
25x
 2x 2y
3
10x
Unless otherwise stated, when simplifying expressions using variables, we must use
absolute value signs.
n
a n  a when n is even.
Examples:
36m 2  6 m
4x 6  2  2  x 3  x 3  2 x 3
3
a 3b 6  3 a  a  a  b 2  b 2  b 2  ab 2
NOTE:
No absolute value signs are needed when finding cube roots, because a real number
has just one cube root. The cube root of a positive number is positive. The cube
root of a negative number is negative.
Class Discussion:
Why must absolute value signs be used for some roots but not all Use the example
below to explain your answer.
Evaluate
More Examples:
x 2 when x = 4 and x = -4 and
3
x 3 when x = 8 and x = -8.
Absolute Value signs are NOT needed here because the index, n, is odd.
Application/Critical Thinking…
A.
The formula for the volume of a sphere is
. Find the radius, to the
nearest hundredth, of a sphere with a volume of 15 in.3.
B.
A student visiting the Sears Tower Skydeck is 1353 feet above the ground.
Find the distance the student can see to the horizon. Use the formula
to approximate the distance d in miles to the horizon when h is
the height of the viewer’s eyes above the ground in feet. Round to the
nearest mile.
. C.
A square garden plot has an area of 24 ft2.
a. Find the length of each side in simplest radical form.
b. Calculate the length of each side to the nearest tenth of a foot.
Application Solutions:
A
ANS:
15 
3
4
3
r 3
15 
4
45
3 4
3
 r 
4 3

 r 3
4
11.25  3.14r 3
r 
3
11.25
3.14
r  1.53in
B.
ANS:
d  1.5h
d  (1.5)(1353)
d  2029.5
d  45miles
C.
ANS:
A  s2
24  s 2
s  24
a) S = 2 6 ;
b) 4.9 ft