7. Algebra Foundations
Middle SchoolAlgebra replaces specific numbers with letters so we can state and solve whole families of problems at once. This chapter covers variables, expressions, the laws of exponents, and the twin skills of expanding and factoring.
Variables, expressions and the laws of indices
A variable is a symbol standing for an unknown or varying number. A term is a number and variables multiplied together (e.g. 3x²y); an expression is terms added or subtracted. Like terms share the same variable part and can be combined: 5x + 2x = 7x.
Exponents obey a fixed set of index laws. For any base a ≠ 0 and exponents m, n:
(am)n = amn (ab)n = anbn
a0 = 1 a−n = 1 / an a1/n = n√a
The distributive law drives two inverse operations: expanding multiplies out brackets, and factoring writes an expression as a product. Key identities:
(a − b)² = a² − 2ab + b²
a² − b² = (a + b)(a − b)
Why letters are a superpower
A number answers one question; a formula answers infinitely many. Writing "area = l × w" captures every rectangle that will ever exist. That is the leap algebra gives you: from solving a problem to solving its whole species.
The index laws are not arbitrary — they follow from what powers mean. a³ × a² is (a·a·a)(a·a), which is five a's multiplied, i.e. a⁵ — so exponents add. Extending the same logic forces the odd-looking rules: a⁰ must equal 1 so that aⁿ/aⁿ = a⁰ stays consistent, and negative exponents must mean reciprocals. Factoring is "un-multiplying," and it matters because a product equal to zero tells you something powerful: at least one factor must be zero — the key that unlocks equation solving.
Worked example: simplify and factor
Simplify (6x³y²) / (2x y⁵) × x²y³.
- Handle the fraction: numbers 6/2 = 3; x³/x = x²; y²/y⁵ = y⁻³. So the fraction is 3x²y⁻³.
- Multiply by x²y³: add exponents — 3 · x²⁺² · y⁻³⁺³ = 3x⁴y⁰.
- Since y⁰ = 1, the result is 3x⁴.
Factor 2x² − 18 fully.
- Take out the common factor 2: 2(x² − 9).
- Recognise the difference of two squares: x² − 9 = (x + 3)(x − 3).
- Full factorisation: 2(x + 3)(x − 3).
Test your understanding
Why does a⁰ = 1 rather than 0?
Use the division law: aⁿ / aⁿ = aⁿ⁻ⁿ = a⁰. But any non-zero quantity divided by itself is 1. So consistency demands a⁰ = 1 for every a ≠ 0. (The case 0⁰ is a special one, often left undefined or context-defined.)
Is (a + b)² equal to a² + b²?
No — this is the most common algebra error. Expanding properly, (a + b)² = (a + b)(a + b) = a² + 2ab + b². The extra 2ab term is real: test a = 3, b = 4, giving 7² = 49, while a² + b² = 25. The difference, 24, is exactly 2ab.
What does a fractional exponent like 8^(2/3) mean, and what is its value?
The denominator is a root, the numerator is a power: 8^(2/3) = (³√8)² = 2² = 4. Equivalently (8²)^(1/3) = ³√64 = 4 — order does not matter. Fractional exponents unify roots and powers into one system.
Factor x² + 5x + 6.
Find two numbers multiplying to +6 and adding to +5: those are 2 and 3. So x² + 5x + 6 = (x + 2)(x + 3). Check by expanding: x² + 3x + 2x + 6 = x² + 5x + 6. ✓
How the ideas connect
Every key idea in this chapter, branching from the core concept — use it to see the whole picture at a glance.
The key facts, visualised
Worked problems, step by step
Follow each solution line by line, then try to reproduce it on paper before moving on.
Example 1Simplify 3x + 5x - 2 + 7.
- Combine like terms 3x + 5x = 8x
- Combine constants -2 + 7 = 5
- Result: 8x + 5
Example 2Factor x^2 + 5x + 6.
- Find two numbers multiplying to 6 and adding to 5
- 2 and 3 work
- So x^2 + 5x + 6 = (x + 2)(x + 3)
Now you try
Work each one out first, then tap to reveal the worked answer.