Application of indices, simple indicial equation for SS 1

Class: SS 1
Term: First Term
Week: 4
Age: 15 years
Duration: 60 minutes
Subject: General Mathematics
Curriculum Theme: Number and Numeration
Focal competence: Applying laws of indices to solve practical daily problem
Key competencies/values: Critical Thinking; Respect; Responsibility
Skills:

• Applying laws of indices to solve practical daily problem

Previous Lesson: Indices and Their Laws
Topic: Laws Of Indices: Application Of Indices, Simple Indicial Equation
Subject Matter: Application of indices, simple indicial equation

Specific Objectives

By the end of the lesson, pupils/students should be able to:

Cognitive Domain

• solve simple indicial equations;
• apply laws of indices to solve practical daily problems.

Affective Domain

• appreciate the importance of indices in simplifying mathematical expressions;
• show responsibility in solving indicial equations.

Psychomotor Domain

• demonstrate the steps for solving indicial equations;
• accurately substitute values into indicial equations.

Social Domain

• collaborate with peers to solve indicial equations.

Reference Materials

The following resources were used in planning this lesson:

• 2025 New Revised Senior Secondary Education Curriculum (SSEC)
• Relevant State Unified Scheme of Work
• New General Mathematics for Senior Secondary Schools 1
• The HeadTeacher Scheme of work For The New Revised Senior Secondary Education Curriculum (SSEC)

Instructional Materials

The teacher will teach this lesson with the aid of:

• Repeated multiplication chart.
• Charts showing laws of indices.
• Whiteboard and markers/chalk.

Rationale for the Lesson

This lesson is important as it provides students with the tools to simplify complex mathematical expressions and solve equations involving powers. Understanding indices is foundational for advanced topics in algebra, calculus, and various scientific fields. It also enhances critical thinking skills through problem-solving.

Prerequisite/Previous Knowledge

Students should have a basic understanding of multiplication, division, and the fundamental laws of indices (multiplication, division, power of a power, zero index, negative index, and fractional index).

Lesson Content/Board Summary

Laws Of Indices: Application Of Indices, Simple Indicial Equation

Review of Laws of Indices

Indices provide a compact way of writing repeated multiplication. The basic laws of indices are essential for solving indicial equations.

1. Multiplication Law: When multiplying numbers with the same base, add the powers.

   (a^m times a^n = a^{m+n})

   Example: (2^3 times 2^2 = 2^{3+2} = 2^5 = 32)

2. Division Law: When dividing numbers with the same base, subtract the powers.

   (a^m div a^n = a^{m-n})

   Example: (5^4 div 5^2 = 5^{4-2} = 5^2 = 25)

3. Power Law: When raising a power to another power, multiply the powers.

   ((a^m)^n = a^{mn})

   Example: ((3^2)^3 = 3^{2 times 3} = 3^6 = 729)

4. Zero Index Law: Any non-zero number raised to the power of zero is 1.

   (a^0 = 1), where (a neq 0)

   Example: (100^0 = 1)

5. Negative Index Law: A number raised to a negative power is the reciprocal of the number raised to the positive power.

   (a^{-n} = frac{1}{a^n})

   Example: (4^{-2} = frac{1}{4^2} = frac{1}{16})

6. Fractional Index Law: A fractional index indicates a root.

   (a^{frac{1}{n}} = sqrt[n]{a})

   (a^{frac{m}{n}} = (sqrt[n]{a})^m = sqrt[n]{a^m})

   Example: (8^{frac{1}{3}} = sqrt[3]{8} = 2)

Simple Indicial Equations

An indicial equation is an equation in which the unknown variable appears as an index or exponent.

Principle for Solving Indicial Equations

The main principle for solving simple indicial equations is to make the bases on both sides of the equation equal. If (a^x = a^y), then (x = y), provided (a neq 0, 1, -1).

Steps to Solve Simple Indicial Equations

1. Express both sides of the equation with the same base. This may involve using the laws of indices to rewrite numbers as powers of a common base.
2. Once the bases are equal, equate the exponents (powers).
3. Solve the resulting linear equation for the unknown variable.

Worked Examples

Example 1

Question: Find the value of (x) in the equation (9^x = 81).

Solution:

Step 1: Express both sides with the same base.

We know that (9 = 3^2) and (81 = 9^2 = (3^2)^2 = 3^4).

So, the equation becomes ((3^2)^x = 3^4).

Using the power law ((a^m)^n = a^{mn}), we get (3^{2x} = 3^4).

Step 2: Equate the exponents.

Since the bases are equal (both are 3), we can equate the exponents:

(2x = 4)

Step 3: Solve for (x).

(x = frac{4}{2})

(x = 2)

Answer: (x = 2)

Example 2

Question: Solve for (x) in the equation (2^{x+1} = 16).

Solution:

Step 1: Express both sides with the same base.

We know that (16 = 2 times 2 times 2 times 2 = 2^4).

So, the equation becomes (2^{x+1} = 2^4).

Step 2: Equate the exponents.

Since the bases are equal (both are 2), we can equate the exponents:

(x+1 = 4)

Step 3: Solve for (x).

(x = 4 – 1)

(x = 3)

Answer: (x = 3)

Teaching Methods/Instructional Techniques

Discussion, Demonstration, Guided Practice, Question and Answer, Explanation, Problem Solving, Pair Work, Group Work, Individual Practice.

Instructional Procedures

Step 1: Introduction

Time: 5 minutes

Teaching Skill: Questioning/Review

Teacher’s Activity: The teacher greets the students and reviews the basic laws of indices by asking questions like, “What happens when we multiply numbers with the same base?” and “What is the value of any non-zero number raised to the power of zero?”.

Pupils’ Activity: Pupils respond to the questions, recalling the laws of indices.

Learning Point: Recall of indices laws.

Step 2: Introduction to Indicial Equations

Time: 10 minutes

Teaching Skill: Explanation/Definition

Teacher’s Activity: The teacher introduces the concept of an indicial equation, explaining that it’s an equation where the unknown is in the power. The teacher then states the principle for solving: making the bases equal.

Pupils’ Activity: Pupils listen attentively and ask questions for clarification.

Learning Point: Understanding indicial equations.

Step 3: Demonstrating Base Equalisation

Time: 10 minutes

Teaching Skill: Demonstration/Modelling

Teacher’s Activity: The teacher demonstrates how to express numbers as powers of a common base using examples such as (81 = 3^4) or (16 = 2^4). The teacher explains that this is the first crucial step in solving indicial equations.

Pupils’ Activity: Pupils observe the examples and participate by suggesting common bases.

Learning Point: Principle of equal bases.

Step 4: Solving Simple Indicial Equations (Example 1)

Time: 10 minutes

Teaching Skill: Guided Practice

Teacher’s Activity: The teacher guides students through solving the first example: (9^x = 81). The teacher encourages students to work in groups as per the activity, guiding them step-by-step to express with the same base, equate powers, and solve for (x).

Pupils’ Activity: Pupils work in groups, follow the teacher’s guidance, and solve the equation.

Learning Point: Solving simple equations.

Step 5: Solving Simple Indicial Equations (Example 2)

Time: 8 minutes

Teaching Skill: Problem Solving

Teacher’s Activity: The teacher presents another example, such as (2^{x+1} = 16), and allows students to attempt it individually or in pairs, applying the steps learned. The teacher circulates to provide support and clarifies misconceptions.

Pupils’ Activity: Pupils attempt to solve the new equation, applying the laws and principles discussed.

Learning Point: Applying laws to equations.

Step 6: Further Application of Laws

Time: 5 minutes

Teaching Skill: Reinforcement

Teacher’s Activity: The teacher provides a quick problem that requires a combination of index laws before equating powers, e.g., (4^{x-1} = 8). The teacher guides students to express both sides in base 2, i.e., ((2^2)^{x-1} = 2^3), then solve.

Pupils’ Activity: Pupils work through the problem, applying multiple index laws.

Learning Point: Further equation practice.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. What is an indicial equation?
2. State the main principle for solving indicial equations.
3. Solve for (x) in (5^x = 125).
4. Solve for (y) in (3^{y-2} = 9).

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Indicial equation solutions.

Step 8: Note-Taking

Time: 10 minutes

Teaching Skill: Guided Writing

Teacher’s Activity: The teacher guides pupils/students to copy the essential Board Summary notes, including the laws of indices and worked examples of indicial equations, into their notebooks.

Pupils’ Activity: Pupils/students copy the notes carefully into their notebooks.

Learning Point: Recording lesson notes.

Step 9: Conclusion

Time: 2 minutes

Teaching Skill: Summarisation

Teacher’s Activity: The teacher summarises the lesson by reiterating the importance of knowing the laws of indices and applying the principle of equal bases to solve indicial equations. The teacher encourages students to practice more problems.

Pupils’ Activity: Pupils listen and reflect on the key takeaways from the lesson.

Learning Point: Importance of indices.

Continuous Assessment/Further Study

Type: Homework

Instruction: Solve the following indicial equations for the unknown variable:

1. (4^x = 64)
2. (7^{y+3} = 49)
3. (2^{2x} = 32)
4. (10^{z-1} = 1000)
5. (3^{2m-1} = 27)

Lesson Keywords

• Indices – The power or exponent to which a number, symbol, or expression is raised.
• Exponent – Another term for index or power.
• Base – The number that is multiplied by itself a certain number of times, as indicated by the exponent.
• Indicial Equation – An equation where the unknown variable is in the exponent.
• Laws of Indices – Rules governing operations with exponents.

Differentiation

Support: For students struggling, provide a chart of common powers (e.g., powers of 2, 3, 5) to help them identify common bases more easily. Offer additional one-on-one guidance during group work and simplify the equations further.

Extension: Challenge advanced students with slightly more complex indicial equations involving fractional or negative indices, or equations that require more manipulation of the bases before equating powers, e.g., (2^{x+1} = frac{1}{8}) or (25^x = 5^{x+3}).

Suggested Lesson Videos

Search on YouTube for “Solving Indicial Equations SS1 Mathematics” or “Laws of Indices Application Grade 10 Math”.