# IGCSE Higher: Rules of Indices

## Scheme of work: IGCSE Higher: Year 10: Term 3: Rules of Indices

#### Prerequisite Knowledge

1. Understanding Basic Indices
• Concept: Knowing the basic rules of indices, including multiplication and division of powers with the same base.
• Example: $a^m \times a^n = a^{m+n}, \quad \frac{a^m}{a^n} = a^{m-n}$
2. Understanding Powers and Roots
• Concept: Knowing how to calculate powers and roots of numbers.
• Example: $\sqrt{a} = a^{1/2}, \quad \sqrt[3]{a} = a^{1/3}, \quad a^{0} = 1$

#### Success Criteria

1. Use Index Notation Involving Fractional Powers
• Objective: Students should be able to use index notation to express and simplify expressions with fractional powers.
• Example: $\sqrt{a} = a^{1/2}, \quad \sqrt[3]{a} = a^{1/3}$
2. Use Index Notation Involving Negative Powers
• Objective: Students should be able to use index notation to express and simplify expressions with negative powers.
• Example: $a^{-n} = \frac{1}{a^n}$
3. Use Index Notation Involving Zero Powers
• Objective: Students should be able to use index notation to express and simplify expressions with zero powers.
• Example: $a^0 = 1$
4. Solve Equations Involving Index Notation
• Objective: Students should be able to solve equations that involve index notation with fractional, negative, and zero powers.
• Example: Solve for $$x$$ in the equation $$4^{2x} = 8^{x+1}$$: $4^{2x} = 8^{x+1} \Rightarrow (2^2)^{2x} = (2^3)^{x+1} \Rightarrow 2^{4x} = 2^{3x+3} \Rightarrow 4x = 3x + 3 \Rightarrow x = 3$

#### Key Concepts

1. Use Index Notation Involving Fractional Powers
• Objective: Students should be able to use index notation to express and simplify expressions with fractional powers.
• Example: $\sqrt{a} = a^{1/2}, \quad \sqrt[3]{a} = a^{1/3}$
2. Use Index Notation Involving Negative Powers
• Objective: Students should be able to use index notation to express and simplify expressions with negative powers.
• Example: $a^{-n} = \frac{1}{a^n}$
3. Use Index Notation Involving Zero Powers
• Objective: Students should be able to use index notation to express and simplify expressions with zero powers.
• Example: $a^0 = 1$
4. Solve Equations Involving Index Notation
• Objective: Students should be able to solve equations that involve index notation with fractional, negative, and zero powers.
• Example: Solve for $$x$$ in the equation $$4^{2x} = 8^{x+1}$$: $4^{2x} = 8^{x+1} \Rightarrow (2^2)^{2x} = (2^3)^{x+1} \Rightarrow 2^{4x} = 2^{3x+3} \Rightarrow 4x = 3x + 3 \Rightarrow x = 3$

#### Common Misconceptions

1. Understanding Fractional Powers
• Concept: Recognizing that fractional powers represent roots of numbers.
• Example: $\sqrt{a} = a^{1/2}, \quad \sqrt[3]{a} = a^{1/3}$
2. Understanding Negative Powers
• Concept: Recognizing that negative powers represent reciprocals of the base raised to the positive power.
• Example: $a^{-n} = \frac{1}{a^n}$
3. Understanding Zero Powers
• Concept: Recognizing that any non-zero number raised to the power of zero is 1.
• Example: $a^0 = 1$
4. Solving Equations with Index Notation
• Concept: Understanding how to manipulate and solve equations that involve indices, including using the laws of indices to simplify and solve.
• Example: Solve for $$x$$ in the equation $$4^{2x} = 8^{x+1}$$: $4^{2x} = 8^{x+1} \Rightarrow (2^2)^{2x} = (2^3)^{x+1} \Rightarrow 2^{4x} = 2^{3x+3} \Rightarrow 4x = 3x + 3 \Rightarrow x = 3$

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