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

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

§ 7.3 Simplifying Radicals

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.

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:

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

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.

Product Rule for Radicals Examples:

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

Quotient Rule for Radicals Examples: