4.1 Laws of Indices
An index expression is written in the form \(a^n\), where \(a\) is the base and \(n\) is the index (or exponent). By applying the laws of indices and properties of algebra, complex algebraic expressions can be simplified, identities can be proven, and exponential equations can be solved accurately.
Learning Objectives
- Simplify algebraic expressions involving positive, negative, and fractional indices using the laws of indices.
- Prove algebraic identities and divisibility of index expressions.
- Solve equations involving indices by expressing both sides in the same base or comparing identical indices.
- Solve real-world contextual problems modelled by exponential and index equations.
Flowchart for Solving Equations Involving Indices
A step-by-step decision pathway for solving any index equation encountered in Form 4 Additional Mathematics.
Same base?
Can both sides be expressed with the same base?
Rewrite as \(a^m = a^n\)
Equate powers: \(m = n\)
Solve for variable.
Same index?
Do both sides have the same index?
Rewrite as \(a^m = b^m\)
Equate bases: \(a = b\)
Solve for variable.
Terms added or subtracted?
Does the equation contain terms added/subtracted?
Factor out the common exponential factor first.
Short Explanation
Apply index laws to simplify expressions and solve equations accurately.
1. Fundamental Laws of Indices
For any non-zero real base \(a\) and \(b\), and rational numbers \(m\) and \(n\): \(a^m \times a^n = a^{m+n}\), \(a^m \div a^n = a^{m-n}\), \((a^m)^n = a^{mn}\), \((ab)^n = a^n b^n\), \(\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}\).
2. Special Index Forms
\(a^0 = 1\) where \(a \neq 0\); \(a^{-n} = \dfrac{1}{a^n}\); \(a^{\frac{1}{n}} = \sqrt[n]{a}\); \(a^{\frac{m}{n}} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}\), where \(a > 0\) and \(n\) is a positive integer.
3. Simplifying Algebraic Expressions
Apply index laws to combine powers with identical bases. Group numerical coefficients and multiply/divide them directly. Factorise common terms when addition or subtraction separates terms containing powers.
4. Proving Divisibility and Identities
Factor out the common term to show divisibility. Use properties of powers such as \(7^{2x-1} = \dfrac{7^{2x}}{7^1} = \dfrac{(7^2)^x}{7} = \dfrac{49^x}{7}\).
5. Solving Equations Involving Indices
Use the same base principle: if \(a^m = a^n\), then \(m = n\) for \(a > 0, a \neq 1\). Use the same index principle: if \(a^m = b^m\), then \(a = b\) for \(a > 0, b > 0\).
6. Real-World Applications
Growth and decay situations can be represented by exponential formulae such as \(J = p(1 + k)^n\), \(h = 10 \times (0.9)^l\), and \(N = 300(3^t)\).
| Law / Property | Algebraic Expression | Condition |
|---|---|---|
| Multiplication law | \(a^m \times a^n = a^{m+n}\) | Same base \(a\) |
| Division law | \(\dfrac{a^m}{a^n} = a^{m-n}\) | Same base \(a\), \(a \neq 0\) |
| Power of a power | \((a^m)^n = a^{mn}\) | — |
| Power of a product | \((ab)^n = a^n b^n\) | — |
| Power of a quotient | \(\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}\) | \(b \neq 0\) |
| Zero index | \(a^0 = 1\) | \(a \neq 0\) |
| Negative index | \(a^{-n} = \dfrac{1}{a^n}\) | \(a \neq 0\) |
| Fractional index | \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\) | \(a > 0\), \(n\) positive integer |
Steps for Simplifying Index Expressions
Apply index laws to combine powers with identical bases.
Group numerical coefficients and multiply/divide them directly.
Factorise common terms when addition or subtraction separates terms.
Expand brackets and convert radical forms into fractional indices.
Same Base Principle
\(\text{If } a^m = a^n, \text{ then } m = n\)For \(a > 0, a \neq 1\).
Same Index Principle
\(\text{If } a^m = b^m, \text{ then } a = b\)For \(a > 0, b > 0\).
Try to Answer First
Answer in your mind, then press "Check Answer".
Simplify the following algebraic expression: \(\dfrac{3^{m+2} - 3^m}{3^m}\).
Solve the equation: \(32^x = \dfrac{1}{8^{x-1}}\).
Show that \(4^{x+2} + 4^{x+1} + 4^x\) is divisible by \(7\) for all positive integers \(x\).
Common Mistakes
- Incorrectly applying product law to addition/subtraction: writing \(3^{x+2} - 3^x\) as \(3^{(x+2)-x} = 3^2\). Correction: Index laws apply to multiplication and division, not addition or subtraction. You must factorise: \(3^x(3^2 - 1)\).
- Sign errors when bringing denominators to numerator: writing \(\dfrac{1}{8^{x-1}}\) as \(8^{-x-1}\). Correction: The entire exponent changes sign: \(8^{-(x-1)} = 8^{-x+1}\).
- Forgetting to distribute powers to all coefficients: expanding \((5x^{-1})^3\) as \(5x^{-3}\) instead of \(5^3 x^{-3} = 125x^{-3}\).
- Multiplying different bases before indices: writing \(2 \times 3^x\) as \(6^x\). Correction: The base \(3\) has power \(x\), while \(2\) has power \(1\); they cannot be combined into base \(6\).
Concept Misunderstandings
Thinking that \(a^0 = 0\).
Any non-zero base raised to the power of zero equals \(1\), i.e., \(a^0 = 1\). For example, \(3^0 = 1\) and \((xy)^0 = 1\).
Believing that \(a^{-n}\) is a negative number.
A negative index represents a reciprocal, not a negative value: \(a^{-n} = \dfrac{1}{a^n}\). For example, \(2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8} > 0\).
Assuming \(a^{\frac{m}{n}} = \dfrac{a^m}{n}\).
A fractional index denotes a root: \(a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m\).
Summary
- Index Laws: \(a^m \times a^n = a^{m+n}\), \(\dfrac{a^m}{a^n} = a^{m-n}\), \((a^m)^n = a^{mn}\), \((ab)^n = a^n b^n\), \(\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}\).
- Special Forms: \(a^0 = 1\), \(a^{-n} = \dfrac{1}{a^n}\), \(a^{\frac{1}{n}} = \sqrt[n]{a}\), \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\).
- Simplification Strategy: Factorise when terms are added/subtracted; apply index laws when terms are multiplied/divided.
- Solving Equations: If \(a^m = a^n\) (where \(a > 0, a \neq 1\)), then \(m = n\). If \(a^m = b^m\) (where \(a > 0, b > 0\)), then \(a = b\). If the equation equals \(1\), rewrite \(1\) as \(a^0\).
Short Activity
Objective: Test your understanding of index laws, simplification and solving index equations.
A. Objective Quiz
1 What is the simplified form of \((5x^{-1})^3 \times 4xy^2 \div (xy)^{-4}\)?
2 Solve the equation \(27(81^{3x}) = 1\).
B. Fill in the Blanks
3 If \(a^m = a^n\) for \(a > 0\) and \(a \neq 1\), then \(m\) is equal to ________.
4 When the expression \(3^{x+4} + 3^{x+5} + 3^x\) is factorised into \(3^x(k)\), the integer value of \(k\) that proves divisibility by 25 is ________.
C. Matching / Drag and Drop
Match each index law or rule to its corresponding algebraic expression. On a phone, tap a term first, then tap the matching box.
