Lesson Notes By Weeks and Term v4 - SHS 1
NUMBER AND ALGEBRAIC PATTERNS
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NUMBER AND ALGEBRAIC PATTERNS
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Subject: Additional Mathematics
Class: SHS 1
Term: 1st Term
Week: 7
Grade code: 1.1.1.LI.5
Strand code: 1
Sub-strand code: 1
Content standard code: 1.1.1.CS.1
Indicator code: 1.1.1.LI.5
Theme: MODELLING WITH ALGEBRA
Subtheme: NUMBER AND ALGEBRAIC PATTERNS
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This lesson introduces learners to indicial equations, which are equations where the unknown variable is in the exponent (or index). This is a powerful concept that builds directly on the laws of indices and skills in solving linear equations. Understanding how to solve these equations is crucial for modelling real-world situations involving rapid growth or decay, such as population growth, compound interest in banking (e.g., at GCB or Ecobank), or even the spread of information on social media platforms like WhatsApp.
I. Recap: The Laws of Indices Before we solve equations, let's refresh our memory on the fundamental laws of indices. For any non-zero base 'a' and rational numbers 'm' and 'n': Multiplication Law: `a^m × a^n = a^(m+n)` (When multiplying powers with the same base, add the indices). Division Law: `a^m ÷ a^n = a^(m-n)` (When dividing powers with the same base, subtract the indices). Power Law: `(a^m)^n = a^(m×n)` (When raising a power to another power, multiply the indices). Zero Index: `a^0 = 1` (Any non-zero number raised to the power of zero is 1). Negative Index: `a^(-n) = 1 / a^n` (A negative index means take the reciprocal). Fractional Index: `a^(m/n) = (n√a)^m` (The denominator is the root, the numerator is the power). II. What is an Indicial Equation? An indicial equation (or exponential equation) is an equation where the variable we are trying to find is located in the index or exponent. Example: In `2^x = 8`, the variable is `x`, which is an index. This is an indicial equation. Non-example: In `x^2 = 9`, the variable is `x`, which is the base. This is a polynomial (quadratic) equation, not an indicial equation. III. The Fundamental Principle for Solving Indicial Equations The primary method we will use is based on a simple, powerful idea:
> If two powers with the same base are equal, then their indices must also be equal. > Mathematically: If `a^x = a^y`, then `x = y` (provided `a > 0` and `a ≠ 1`).
Method for Solving Simple Indicial Equations: Manipulate the Equation: Rewrite the terms on both sides of the equation so they have the same base. Equate the Indices: Once the bases are the same, set the indices equal to each other. Solve the Resulting Equation: Solve the new (usually linear) equation for the variable.