Sample · 01-M1-ALGEBRA and FUNCTIONS
Chapter 1 Overview
TMUA Chapter 1: M1-ALGEBRA and FUNCTIONS
This chapter is written for a TMUA Mathematics book. All topic explanations, worked examples, and multiple-choice practice are collected under one chapter: M1-ALGEBRA and FUNCTIONS. The chapter builds from indices into surds and radicals, then adds rational expressions, rational inequalities, ratio and proportion before continuing through quadratics, simultaneous equations, inequalities, polynomials, and polynomial theorems. Every MCQ is designed for non-calculator reasoning, with full method, Trap Analysis, Teacher's Note, and EduCoach Note.
📋 How to Use This Chapter
Read each short lesson first, study the worked example, then attempt the multiple-choice questions. Open the solution only after choosing an answer. The mathematics is written in textbook display style so that expressions, fractions, roots, and exponents are readable on screen and in print.
📋 TMUA Chapter Map
| TMUA Chapter | Topic sequence inside this chapter |
|---|---|
| M1-ALGEBRA and FUNCTIONS | Indices; Surds and Radicals; Rational Expressions, Rational Inequalities, Ratio and Proportion; Quadratic Functions; Simultaneous Equations; Inequalities; Polynomials including Advanced Polynomial Examples and Polynomial Functions and Graphs; Polynomial Theorems |
| Exam style | Non-calculator MCQ reasoning with exact arithmetic and algebraic structure |
Indices
Indices
Indices are a compact language for repeated multiplication, roots, reciprocals, and scaling. In TMUA-style questions, index laws are often tested indirectly: a question may look like algebra, simultaneous equations, or functions, but the key move is usually to rewrite every expression using a common base or a rational exponent.
The safest strategy is to identify the base, convert roots to fractional powers, and then apply the laws of indices only to multiplication, division, and powers of powers. Do not apply index laws across addition or subtraction.
📋 Core index laws
| Law | Formula | Condition or note |
|---|---|---|
| Product | \(a^m a^n=a^{m+n}\) | Same base |
| Quotient | \(\dfrac{a^m}{a^n}=a^{m-n}\) | Same non-zero base |
| Power of a power | \((a^m)^n=a^{mn}\) | Multiply exponents |
| Negative power | \(a^{-n}=\dfrac{1}{a^n}\) | \(a\ne 0\) |
| Zero power | \(a^0=1\) | \(a\ne 0\) |
| Rational power | \(a^{m/n}=\sqrt[n]{a^m}\) | Use real-domain care when \(n\) is even |
⚠️ High-frequency traps
- \(a^m+a^n\ne a^{m+n}\) in general.
- \((ab)^n=a^n b^n\), but \((a+b)^n\ne a^n+b^n\) in general.
- If \(0<a<1\), increasing the exponent makes the value smaller.
- For even roots, domain restrictions matter: \(a^{1/2}\) requires \(a\ge 0\) in real arithmetic.
Medium Example 1 — using a common base
Question: If \(2^a=x\), express \(8^{a+1}\) in terms of \(x\).
Working: Since \(8=2^3\), \(8^{a+1}=(2^3)^{a+1}=2^{3a+3}=8(2^a)^3=8x^3\).
Medium Example 2 — fractional exponent form
Question: Simplify \(\sqrt[3]{x^5}\cdot\sqrt{x}\), assuming \(x>0\).
Working: Convert to exponents: \(x^{5/3}x^{1/2}=x^{10/6+3/6}=x^{13/6}\).
Rational Exponents
A rational exponent has the form \(m/n\). The denominator represents the root degree and the numerator represents the power: \(a^{m/n}=\sqrt[n]{a^m}\). This notation is especially useful for comparing quantities, simplifying mixed radicals, and converting nested roots into a single power.
For TMUA questions, rational exponent form is usually faster than radical form because it allows direct addition, subtraction, and multiplication of exponents.
Continue reading — 18 chapters, worked examples and practice.
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