This document discusses roots and radicals. It defines that for every power, there is a corresponding root. It explains that the nth root of a number a is the number that when raised to the nth power equals a. It discusses that if n is odd there is one real nth root, and if n is even and the number is positive there are two real nth roots, and if the number is negative there are no real nth roots. It provides examples of finding cube and fourth roots. It also discusses simplifying radicals and working with variables in radicals.

1. 6.1 Roots and Radical
Expressions

2. Powers and Roots
• Corresponding to every power there is a root.
– Examples:
If , then
If , then
If , then

3. The nth Root
• If , with a and b real numbers and n a
positive integer, then a is an nth root of b
If n is odd… If n is even…
There is one real nth root of b And b is positive, there are
two ( + )real nth roots of b.
• The positive root is the principle
root:
• The negative root is its
opposite:
And b is negative, there are no
real nth roots of b.

4. Vocabulary
• You use a radical sign to indicate a root
• The number or term under the radical sign is
called the radicand
• The index gives the degree of the root.
n
a

5. Finding all Roots
• To find the roots of a number, ask yourself
“What raised to the n power, gives the
radicand?”
– To find the roots of a fraction, consider the
numerator and denominator separately
– Don’t forget about Positive and Negative numbers

6. Example: Find all the real roots
Find all the real cube roots of each number
1. 1000
2. 0.008
3.
1
27

7. Example: Find all the real roots
Find all the real fourth roots of each number
1. 1
2. -0.0001
3.
16
81

8. Example: Find each Root
1. 3
8
4
2. 1
2
3. 2

9. nth Roots of nth Powers
n n
a if n is odd
For any real number a, a
a if n is even
It is important to remember this rule,
especially when working with variables!

10. Simplifying Radicals
• Break the radical down into perfect powers of
n.
– Remember Exponent Rules!

11. Example: Simplify
4
1. 81x
2. 16x 8
3 6 9
3. ab

12. Example: Simplify
3 12 15
1. a b
8 12
2. 4
x y
4 12 16
3. x y

13. Example (p364)
Some teachers adjust test scores when a test
is difficult. One teacher’s formula for adjusting
scores is A 10 R where A is the adjusted
score and R is the raw score. If the raw scores
on one test range from 36 to 90, what is the
range of adjusted scores?

14. Homework
• P364 #1 – 30 all, 37