Description

In this topic, we turn our attention to writing radicals and roots to rational exponents. Then, using the laws of exponents, we extend our knowledge of simplifying radicals. We further investigae arithmetic operations involving radicals, division of radicals including rationalize the denominator; solving equations involving radicals; explore in more detail complex numbers and where to apply these skills to real world scenarios.

Sections of this topic include:

1. Rational Exponents

   We begin exploring this section with the notion of rational exponents. Groups within this section include: simplifying expressions with rational exponents; simplifying variable expressions with rational exponents; factoring with rational exponents; simplifying larger expressions with rational exponents; simplifying expressions with variable rational exponents; and evaluating numerical expressions with rational exponents using a calculator.

2. Radicals and Rational Exponents

   We next investigate the relationship between expressions with radicals and with rational exponents. Groups within this section include: writing in radical notation; writing radicals as expressions with rational exponents; simplifying radicals that contain perfect squares; simplifying radicals that are not perfect squares; evaluating square root functions; simplifying radicals with non-negative expressions; and evaluating radicals using calculators.

3. Arithmetic Operations Involving Radicals

   We are now interesting in investigating how to add or subtract like radicals (called combining like radicals). Groups within this section include: combining like radicals; combining like radicals with simplifying radicals; factoring radicals out of expressions; writing radicals in simplest radical form; and determining if radical solutions satisfy equations.

4. Division by Radicals; Rationalizing

   In this section, we investigate how to divide radicals including removing radicals from denominators using the process known as Rationalizing the Denominator. Groups within this section include: rationalizing the denominator with single terms in the denominator; and adding subtracting and dividing functions with rationalizing the dnominator.

5. Solving Equations Involving Radicals

   Combining our skills of solving equations with radicals, we explore how to solve equations involving radicals. The group within this section is called: solving equations involving radicals.

6. Complex Numbers

   In the 1500's an Italian mathematician by the name of Gerolamo Cardano first discovered imaginary and complex numbers. Since then, their application to most fields of mathematics, science and engineering has become a very helpful tool in investingating problems. Here, we investigate what complex numbers are as well as explore how to perform arithmetic operations that contain imaginary or complex numbers. Groups within this section include: simplifying powers of i; simplifying complex numbers; simplifying radicals containing negative numbers; solving equations containing complex numbers; expanding complex numbers raised to a power; and evaluating functions with complex numbers.

7. More Applications

   This section investigates real world applications using radicals, rational exponents, and complex numbers.