Math Lesson 3:
 Exponents and Surds

3.1 The Number System

This section will cover the fundamental concepts of the number system, including natural numbers, integers, rational numbers, and irrational numbers. We will explore how these different types of numbers are related and how they are represented on the number line.

Exercise: Number Classification

Classify the following numbers as natural, integer, rational, or irrational:

  • 5
  • -3
  • 2/3
  • √2
  • π

3.2 Working with Irrational Numbers

This section will delve into the properties of irrational numbers, specifically square roots. We will learn how to simplify surds, perform operations (addition, subtraction, multiplication, and division) with surds, and rationalize denominators.

Exercise: Simplifying Surds

Simplify the following surds:

  • √12
  • √27
  • √48

3.3 Exponents

This section will introduce the concept of exponents and the laws of exponents. We will learn how to work with exponents in various situations, including multiplication, division, raising to a power, and negative exponents.

Exercise: Exponent Laws

Simplify the following expressions using the laws of exponents:

  • x3 * x5
  • y8 / y2
  • (z4)2

3.4 Exponential Equations

This section will explore how to solve equations involving exponents. We will learn different methods for solving these equations, including isolating the exponential term, using logarithms, and applying the properties of exponents.

Exercise: Exponential Equation

Solve the following exponential equation for x:

2x = 16

3.5 Equations with Rational Exponents

This section will cover equations with rational exponents (fractional exponents). We will learn how to solve these equations using various methods, including isolating the term with the rational exponent, raising both sides of the equation to the reciprocal of the exponent, and simplifying the result.

Exercise: Rational Exponent Equation

Solve the following equation for x:

x3/2 = 27

3.6 Exam Type Examples

This section will provide a collection of exam-style problems related to exponents and surds. These problems will test your understanding of the concepts covered in the previous sections and prepare you for real-world applications of these topics.

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