Lesson Notes By Weeks and Term v4 - SHS 1

Lesson Video

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Performance objectives

• Define indices and explain their laws.
• Solve simple equations involving indices.
• Apply the laws of indices to simplify expressions.
• Work together to solve problems involving indices.

Lesson summary

This lesson introduces learners to indicial equations, which are algebraic equations where the unknown variable appears as an exponent or index. Mastering this topic is fundamental as it forms the basis for understanding exponential growth and decay phenomena. These concepts are visible all around us in Ghana – from calculating compound interest on savings at a bank like GCB or Ecobank, to modelling the population growth of our cities like Accra and Kumasi, and even understanding scientific concepts like the half-life of materials. By learning to solve these equations, you will gain powerful tools for modelling and solving real-world problems.

Lesson notes

What is an Indicial Equation? An indicial equation (or exponential equation) is an equation in which the variable we are trying to find is in the index or power. For example: `2^x = 8` is an indicial equation because the unknown is `x`, which is a power. `x^2 = 9` is a not an indicial equation; it is a quadratic equation because the unknown `x` is the base. Recap: The Laws of Indices Before we can solve indicial equations, we must be experts in using the laws of indices. Let's review them quickly. For any non-zero base `a` and integers `m` and `n`:

| Law | Formula | Example | | :-- | :--- | :--- | | 1. Multiplication Law | `a^m × a^n = a^(m+n)` | `2^3 × 2^4 = 2^(3+4) = 2^7` | | 2. Division Law | `a^m ÷ a^n = a^(m-n)` | `5^6 ÷ 5^2 = 5^(6-2) = 5^4` | | 3. Power of a Power Law| `(a^m)^n = a^(mn)` | `(3^2)^5 = 3^(2×5) = 3^10` | | 4. Zero Index Law | `a^0 = 1` | `1,000,000^0 = 1` | | 5. Negative Index Law| `a^-n = 1 / a^n` | `7^-2 = 1 / 7^2 = 1/49` | | 6. Fractional Index Law| `a^(1/n) = n√a` | `64^(1/3) = ³√64 = 4` | | 7. General Fractional Law| `a^(m/n) = (n√a)^m` | `8^(2/3) = (³√8)^2 = 2^2 = 4` | The Golden Rule for Solving Indicial Equations The most important principle for solving these equations is this:

> If `a^x = a^y`, then it must be true that `x = y`. > (This rule applies when the base `a` is a positive number and `a ≠ 1`).

This means our main strategy is to manipulate the equation until we have the same base on both sides. Once we achieve this, we can simply equate the powers and solve the resulting (usually linear) equation. Step-by-Step Method and Worked Examples

Evaluation guide

• Pose and solve simple equations involving indices.
• Collaborative Learning, Talk for Learning, Building on what others say.
• Solving simple indicial equations.
• Learners in their groups apply their knowledge in solving linear equations and indices to solve