1. Roots, Radicals, and Complex Numbers
Chapter 7
Roots, Radicals, and Complex Numbers

2. Chapter Sections Objectives: 1. Find the nth root of a number.
2. Approximate roots using a calculator.
3. Simplify radical expressions.
4. Evaluate radical functions.
5. Find the domain of radical functions.
6. Solve applications involving radical functions.
7.1 – Roots and Radicals
7.2 – Rational Exponents
7.3 – Simplifying Radicals
7.4 – Adding, Subtracting, and Multiplying Radicals
7.5 – Dividing Radicals
7.6 – Solving Radical Equations
7.7 – Complex Numbers
Chapter 1 Outline

3. Find Square Roots
A square root is a radical expression that has an index of 2. The index of a square root is generally not written. Thus, 𝑥 means 2 𝑥 . In general the words “square root” , refer to to the principal or positive square root.

4. Example 1: Evaluate each root, if possible. a. b. c.
is not a real number

5. Find Cube Roots
The cube root of a number a, written 𝟑 𝒙 , is the number b such that b3 = x.
Example 2:

6. Example 3:
Approximate the roots using a calculator or table in the endpapers. Round to three decimal places. a. b. c.
Some roots, like are called irrational because we cannot express their exact value using rational numbers. In fact, writing with the radical sign is the only way we can express its exact value.

7. Example 4: For each function, find the indicated value(s).
Radical function: A function containing a radical expression whose radicand has a variable.
Example 4: For each function, find the indicated value(s).

8. Understand Odd, Even & nth Roots
Odd Root
Even Root
The nth root of a, 𝑛 𝑥 , where n is an odd index and a is any real number, is called an odd root and is the real number b such that bn = a.
The nth root of a, written 𝑛 𝑥 , where n is an even index and a is a nonnegative real number, is called an even root and is the nonnegative real number b such that bn = a.
Example 5:

9. Example 6:
Find each value.

10. Example 7:
Find the root. Assume variables represent nonnegative values. a. b. c. d. e.

11. Expressions of the Form
It is tempting to write but the next example shows that, as a rule, this is untrue.
For any real number a,
(The principal square root of a2 is the absolute value of a.)

12. Evaluate Radicals Using Absolute Value
Example 8:
Find the root. Assume variables represent any real number. a. b. c. d. e. f.

13. Example 9: The index is even, the radicand must be nonnegative.
Find the domain of each of the following. a. b. c.
The index is even, the radicand must be nonnegative.
Conclusion: The domain of a radical function are as follows:
With an Even index: Values that keep its radicand nonnegative.
With an Odd index: All real numbers

14. Example 10: Given: d= 800 feet, t=?
If you drop an object, the time (t) it takes in seconds to fall d feet is given by Find the time it takes for an object to fall 800 feet.
Given: d= 800 feet, t=?