Lesson 5: Indices and Laws of Exponents

👨🏽‍🏫 Introduction

Have you ever seen something like and wondered what it really means? That’s called an exponent (or index). It tells us how many times to multiply a number by itself. 🔁

Understanding indices is important in math, science, and computing. They help us write very large or very small numbers in compact form.

In this lesson, you’ll learn to:

• Understand the meaning of indices (exponents)
• Apply the laws of indices to simplify expressions
• Use zero and negative indices
• Work with fractional indices (briefly introduced)

🧠 Key Concepts

📌 1. What Is an Index (Exponent)?

When we write , it means:

• The big number (2) is the base
• The small number (3) is the index or exponent

📏 2. Laws of Indices

Here are the key rules to remember when working with the same base:

🔸 Law 1: Product of Powers

Rule:

Example:

🔹 Law 2: Quotient of Powers

Rule:

Example:

🔸 Law 3: Power of a Power

Rule:

Example:

🔹 Law 4: Zero Index

Rule: (for any non-zero a)

Example:

🔸 Law 5: Negative Indices

Rule:

Example:

🔹 Law 6: Fractional Indices (Intro Only)

Rule:

Example:

✍🏽 Sample Problem Walkthroughs

1. Simplify:
   
2. Simplify:
   
3. Evaluate:
   
4. Convert:
   
5. Simplify:
   

🧪 Practice Exercises

✏️ A. Simplify the following:

✏️ B. Evaluate:

✅ Answers with Explanations

• A1:
• A2:
• A3:

• B1:
• B2:
• B3:

📌 Recap

• ✅ Indices tell us how many times to multiply a number by itself
• ✅ Laws of indices help us simplify large expressions
• ✅ Zero and negative exponents follow their own special rules
• ✅ Fractional indices represent roots (like square roots)

💭 Reflection Prompt

Think about scientific values like mass of atoms or speed of light. Why is using indices more helpful than writing out all the zeros? Try writing a large or small number both in index form and in full.