1. Roots and Radical Expressions
What you’ll learn
To find nth roots. Add, subtract, multiply, and divide radical expressions.
Vocabulary
nth root, principal root,
radicand, index

2. Take a note: Corresponding to every power, there is a root.
Example:
As there are squares(second powers), there are square roots.
As there are cubes (third powers), there are cube roots, and so on.
index
radicand
Type of number b
Number of Real nth
Roots When n is Odd
Roots When n is Even
positive 1 2
negative
none
Radical sign

3. Problem 1: Finding all Real roots
n is 3 and 3 is odd.
So there is only
one real cube root
of a number.
Answers:
n=4; even. Since 1 is positive, there are two real fourth roots
Answers:

4. a)What are the real fifth roots of 0,-1 and 32?
n=4; even. Since
is negative, there are no
real fourth roots of
Answer:
No real fourth roots
n=4; even. Since 16/81 is
positive , there are two real
fourth roots.
Answer
Your turn
a)What are the real fifth roots of 0,-1 and 32?
b) What ere the square roots of0.01, -1 and ?
Answers

5. Problem 2: Finding Real Roots.
There is no real root because there is no real number whose fourth power is -1
It is tempting to conclude that
,but the above problem shows
that is not the case If n is even, then
is positive even if a itself is
negative

6. Your turn Answers: There is no real root because there is no real
number whose fourth
power is -81
There is no real root
because there is no real
number whose square
is -49

7. Remember when the radicand contains a variable
Take a note
Remember when the radicand contains a variable
expression, you MUST include the absolute value when
n is even, and omit it when n is odd.
Problem 3: Simplifying Radical Expressions
The index of the square root is even.
However, X to the power 4 is always nonnegative.
The index is odd, so you can not include absolute values.
The index is even. The absolute value symbols ensure that the root is positive when is negative. Absolute value symbols are not needed for since is always nonnegative..

8. Take a note: nth Root of nth-Powers
When a value has an exponent of n and you take the nth root you will get the value back again ..
... when a is positive (or zero):
(i.e. for a ≥ 0)
Example:
... or when the exponent is odd:
(i.e. when n is odd
Example:
... but when a is negative and the exponent is even you get this:
... so we have:
(when n is even)
(Note: |a| means the absolute value of a, in other words any negative becomes a positive
Example:

9. Your turn
Answers

10. Your turn again
Find the two real solutions for each equation
Answers:

11. Problem 4: Using a Radical Expression
The velocity of a falling object can be found
using the formula where v is the
velocity (in feet per second) and h is the distance the
object has already fallen.
What is the velocity of the object after 10 ft fall?
How much does the velocity increase if the object
falls 20 ft rather than 10 ft
Answers: b) a)

12. Your turn:
The speed s in meters per second of a car leaving a
skid mark d meters long after the brakes are applied
is given by the formula
If you measure a skid mark and it is 180 m long, how fast was the car going before it braked?
Answer: about 52 m/s

13. Classwork odd Homework even
TB pgs Exercises 10-48