Transcript Roots of Real Numbers

Roots of Real Numbers
Roots of Real Numbers

Simplify radicals.

Use a calculator to approximate
radicals.
Real Life Example 1:
OCEANOGRAPHY
The speed of a wave can be estimated using the
formula s  1.34 l , where I is the length of the
wave in feet. This is an example of an equation
that uses a square root.
Key Concepts
The symbol
n
indicates an nth root.
Radical sign
index
n
50
radicand
Some numbers have more than one real nth root.
For example, 36 has two square roots, 6 and -6.
When there is more than one real root, the
nonnegative root is called the principal root.
!
Key Concepts
The chart below gives a summary of the real nth
roots of a number b.
n
Even
n
b
if b >0
if b<0
One positive root,
one negative root
No real roots
 625  5
4
One positive root, no
negative roots
Odd
n
b
3
82
 4 is not a real #
no positive roots,
one negative root
5
 32  2
b=0
One real root, 0
n
0 0
Example 1:
Simplify.
a.)  16x 6
Answers:
 4x 3
(Find the square root of 16, then divide the exponent on the
variable by the index. The index is 2.)
b.)

c.)
5
q
3
 5
4
243a10b15

 q 5
3

2
3a 2b 3
(Find what number can be multiplied by itself 5 times. Then
divide the exponents on each variable.)
d.)
4
 4 is not a real number
Example 2:
REMEMBER: the general rule is to divide the
exponent on the inside by the index!!
Simplify.
a.)
6
t6
t
Since the index is even, the principal root is
nonnegative. Since t could be negative, you
must take the absolute value of t to identify the
principal root.
15
3
5


243
x

2


3
x

2
b.)
Example 3: PHYSICS
The time T in seconds that it takes a pendulum to
make a complete swing back and forth is given by
the formula T  2 Lg , where L is the length of the
pendulum in feet and g is the acceleration due to
gravity, 32 feet per second squared. Find the
value of T for a 1.5 foot-long pendulum.
1.5
T  2
 2(3.14) 0.046875  6.28(0.2165063)
32
Answer: about 1.36 s
 1.36