1. Drill #37 Factor the following polynomials Simplify.
Find the value of r:

2. Drill #38
Factor the following polynomials
Simplify.

3. Drill #60 Factor the following polynomial
Evaluate the following roots:

4. 7-4 Nth Roots
Objective: To simplify radicals having various indices, and to use a calculator to estimate the roots of numbers.

5. Square Roots What power is a square root?
A square is the inverse of a square root…

6. Square Root* Definition: For any real numbers a and b, if
then a is a square root of b or
We can also write square roots using the ½ power.

7. Cube Root* Definition: For any real numbers a and b, if
then a is a cube root of b or
We can also write cube roots using the 1/3 power.

8. nth Root* Definition: For any real numbers a and b, if
then a is a nth root of b or
We can also write nth roots using the power.

9. How many ways can you … Multiply two numbers to get a positive #?
Multiply two numbers to get a negative #?

10. Examples: Roots (of powers of 2)
Even Roots: Odd Roots:

11. Principal Root*
Definition: When there is more than one real root of a number (even numbered roots), the non-negative root is the principal root
 Principal Root
 Negative Root
 Both

12. Examples: Roots (of powers of 2)
Both Principal Negative
Roots: Roots: Roots:

13. Roots of negative numbers*
Even roots: Negative numbers have no even roots. (undefined)
Odd Roots: Negative numbers have negative roots.

14. Examples: Roots (of powers of 2)
Even Roots: Odd Roots:

15. Roots: Number and Types
Even Roots
Odd Roots
Positive
2 (one positive, one negative)
1 (positive)
Negative
0 (undefined)
1 (negative)

16. Even Roots (of variable expressions)*
When evaluating even roots (n is even) use absolute values (if resulting power is odd).

17. Odd Roots (of variable expressions)*
When evaluating odd roots (n is odd) do not use absolute values.

18. Evaluating Roots of Monomials
To evaluate nth roots of monomials:
(where c is the coefficient, and x, y and z are variable expressions) or
Simplify coefficients (if possible)
For variables, evaluate each variable separately

19. Evaluating Roots of Monomials*
To find a root of a monomial
Split the monomial into a product of the factors, and evaluate the root of each factor.
Variables: divide the power by the root Coefficients: re-write the number as a product of prime numbers with powers, then divide the powers by the root.

20. Use a calculator to find the roots*
Find the following roots using a calculator (round to the 3 decimal places):

21. Examples: Simplify

22. Examples: Simplify