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

2. Chapter Sections 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. § 7.3
Simplifying Radicals

4. Understand Perfect Powers
Perfect Power, Perfect Square, Perfect Cube
A perfect power is a number or expression that can be written as an expression raised to a power that is a whole number greater than 1.
A perfect square is a number or expression that can be written as a square of an expression. A perfect square is a perfect second power.
A perfect cube is a number or expression that can be written as a cube of an expression. A perfect cube is a perfect third power.

5. Perfect Powers
This idea can be expanded to perfect powers of a variable for any radicand.
A quick way to determine if a radicand xm is a perfect power for an index is to determine if the exponent m is divisible by the index of the radical.
Since the exponent, 20, is divisible by the index, 5, x20 is a perfect fifth power.
Example:

6. Product Rule for Radicals
For nonnegative real numbers a and b,
Example:
√20 can be factored into any of these forms.

7. Product Rule for Radicals
To Simplify Radicals Using the Product Rule
If the radicand contains a coefficient other than 1, if possible, write it as a product of two numbers, one of which is the largest perfect power for the index.
Write each variable factor as a product of two factors, one of which is the largest perfect power of the variable for the index.
Use the product rule to write the radical expression as a product of radicals. Place all the perfect powers (numbers and variables) under the same radical.
Simplify the radical containing the perfect powers.

8. Product Rule for Radicals
Examples:

9. Quotient Rule for Radicals
For nonnegative real numbers a and b,
Examples:

10. Quotient Rule for Radicals
Examples: