# Algebraic Manipulation & Indices

> Mathematics · CIE IGCSE 0580
> Source: https://www.owlsprep.com/study/cie-0580-u2-algebraic-manipulation-indices/

This guide covers all Core and Extended content for algebraic manipulation and indices in CIE IGCSE Maths 0580, including substitution, bracket expansion, factorisation, index laws, and Extended-only skills like completing the square.

**Prerequisites:** [Basic algebra notation and operations](https://www.owlsprep.com/study/cie-0580-u1-basic-algebra/); [Integer arithmetic rules](https://www.owlsprep.com/study/cie-0580-u1-number-operations/)

## Learning objectives

- Substitute numerical values into algebraic expressions and formulae accurately
- Simplify expressions by collecting like terms
- Expand single brackets and products of two brackets for Core tier
- Factorise expressions by extracting common factors for Core tier
- Apply index laws with positive, zero and negative indices, and solve simple index equations such as $2^x = 32$ (Core tier)
- (Extended only) Expand products of 3+ brackets, use 5 special factorisation forms, complete the square, and use fractional indices

## Substitution & Simplifying Like Terms

Substitution is the process of replacing variables in an expression or formula with given numerical values, then simplifying using BODMAS order of operations. To simplify expressions, you first group and combine like terms.

**Worked example:** Evaluate $2x^2 - 3y + 5$ when $x=2$ and $y=-1$.

1. Substitute the given values for x and y

   $$2(2)^2 - 3(-1) + 5$$
2. Calculate exponents first, then multiply

   $$2(4) + 3 + 5 = 8 + 3 +5$$
3. Add terms to get the final result

   $$16$$

**Like Terms** — Terms that have identical variable parts, e.g. $3x$ and $-7x$ are like terms, but $2x^2$ and $4x$ are not.

*Example:* Simplify $3a + 2b - 5a + 7b$

**Worked example:** Simplify $3a + 2b - 5a + 7b$.

1. Group like terms together, keeping their signs

   $$(3a - 5a) + (2b + 7b)$$
2. Combine coefficients of each group

   $$-2a +9b$$

> **Exam tip:** Always carry the sign of each term when collecting like terms: negative signs apply only to the term immediately after them.

## Expanding Brackets (Core)

Expanding brackets uses the distributive law: every term inside the bracket is multiplied by the term outside the bracket. For products of two brackets, multiply every term in the first bracket by every term in the second.

**Worked example:** Expand $4(2x - 3y)$.

1. Multiply each term inside the bracket by 4

   $$4 \times 2x - 4 \times 3y$$
2. Simplify to get the result

   $$8x - 12y$$

> **FOIL for two brackets**
>
> F = First terms, O = Outer terms, I = Inner terms, L = Last terms

**Worked example:** Expand $(x + 3)(2x - 5)$.

1. Apply FOIL rule

   $$x \times 2x + x \times (-5) + 3 \times 2x + 3 \times (-5)$$
2. Calculate each product

   $$2x^2 -5x +6x -15$$
3. Combine like terms

   $$2x^2 +x -15$$

> **Exam tip:** After expanding pairs of brackets, always simplify by collecting like terms to get full marks.

## Core Factorisation & Integer Index Laws

Core factorisation involves extracting the highest common factor (HCF) of all terms in an expression. Integer index laws are used to simplify expressions with powers of the same base.

**Highest Common Factor (HCF) for algebra** — The largest number and highest power of each variable that divides all terms in the expression.

*Example:* HCF of $6x^2$ and $9x$ is $3x$

**Worked example:** Factorise $12x^3y - 8x^2y^2$.

1. Identify HCF of all terms: 4, $x^2$, $y$ → HCF = $4x^2y$
2. Divide each term by the HCF and write the result in brackets

   $$4x^2y(3x - 2y)$$

1. Multiplication: $a^m \times a^n = a^{m+n}$
2. Division: $a^m \div a^n = a^{m-n}$
3. Power of a power: $(a^m)^n = a^{mn}$
4. Zero index: $a^0 = 1$ for $a \neq 0$
5. Negative index: $a^{-n} = \frac{1}{a^n}$ for $a \neq 0$

**Worked example:** Simplify $\frac{15a^4b^2}{3a^2b} \times (2a^3)^2$.

1. Simplify the fraction first using division law

   $$5a^{2}b \times (2a^3)^2$$
2. Apply power of a power law to the second term

   $$5a^2b \times 4a^6$$
3. Multiply coefficients and apply multiplication law for indices

   $$20a^8b$$

**Worked example:** Solve $3^{2x-1} = 27$ for $x$.

1. Rewrite both sides with the same base: $27 = 3^3$

   $$3^{2x-1} = 3^3$$
2. Equate exponents since the bases are equal

   $$2x - 1 = 3$$
3. Solve for x

   $$2x = 4 → x = 2$$

> **Exam tip:** Always verify your factorised answer by expanding it back out to match the original expression.

## Extended Adds: Advanced Expansion & Factorisation

Extended tier requires expanding products of 3+ brackets, and factorising using 5 special forms: common factor, difference of two squares (DOTS), grouping, $ax^2+bx+c$, and $a^2x^2 - b^2y^2$.

**Worked example:** Expand $(x+2)(x-3)(2x+1)$.

1. Expand first two brackets first

   $$(x^2 - 3x +2x -6)(2x +1) = (x^2 -x -6)(2x+1)$$
2. Multiply each term in the first bracket by each term in the second

   $$x^2(2x+1) -x(2x+1) -6(2x+1)$$
3. Simplify and collect like terms

   $$2x^3 +x^2 -2x^2 -x -12x -6 = 2x^3 -x^2 -13x -6$$

**Worked example:** Factorise $2x^2 -7x -15$.

1. Find factors of $2x^2$ (2x and x) and factors of -15 that sum to -7x when cross-multiplied: +3 and -5
2. Write the factorised form

   $$(2x +3)(x -5)$$
3. Verify by expanding to match original expression

> **Exam tip:** For quadratic factorisation with $a>1$, test pairs of factors for the $x^2$ coefficient and constant term until you get the correct middle term when expanded.

## Extended Adds: Completing the Square & Advanced Indices

Extended additionally requires completing the square for quadratics and applying fractional index laws such as $a^{m/n} = (\sqrt[n]{a})^m$. (Negative indices and simple same-base index equations are Core content, covered above.)

**Worked example:** Complete the square for $2x^2 + 8x - 3$.

1. Factor out the coefficient of $x^2$ from the first two terms

   $$2(x^2 +4x) -3$$
2. Complete the square inside the bracket: $x^2+4x = (x+2)^2 -4$
3. Substitute back and simplify the constant term

   $$2[(x+2)^2 -4] -3 = 2(x+2)^2 -8 -3 = 2(x+2)^2 -11$$

- Fractional index: $a^{1/n} = \sqrt[n]{a}$, $a^{m/n} = (\sqrt[n]{a})^m$

**Worked example:** Evaluate $16^{3/4}$.

1. Take the $n$th root first: $16^{1/4} = \sqrt[4]{16} = 2$

   $$16^{1/4} = 2$$
2. Raise to the numerator power

   $$16^{3/4} = 2^3 = 8$$

## Common pitfalls

- **Wrong:** Forgetting to apply a negative coefficient to all terms when expanding a bracket
  - Why it fails: Negative numbers multiply through every term inside the bracket, not just the first term
  - Correct: Multiply every term inside the bracket by the full coefficient (including its sign) before simplifying
- **Wrong:** Adding exponents when multiplying terms with different bases
  - Why it fails: Index laws only apply to terms with identical bases
  - Correct: Keep terms with different bases separate when simplifying index expressions
- **Wrong:** Trying to factorise a sum of two squares (e.g. $x^2 + 4$) using DOTS rule
  - Why it fails: Sum of two squares has no linear factors at IGCSE level
  - Correct: Only apply DOTS to expressions of the form $a^2 - b^2$
- **Wrong:** Forgetting to adjust the constant term after factoring out the leading coefficient when completing the square
  - Why it fails: Factoring changes the value of the expression, so you need to compensate to keep it equivalent
  - Correct: After completing the square inside the factored bracket, expand and adjust the constant term to match the original expression
- **Wrong:** Treating $a^0$ as 0 instead of 1 for non-zero $a$
  - Why it fails: Any non-zero value raised to the power of 0 equals 1 by definition of index laws
  - Correct: Immediately replace any non-zero base raised to power 0 with 1 when simplifying

## Cheatsheet

| Skill | Tier | Rule/Method |
| --- | --- | --- |
| Substitution | Core | Replace variables with given values, follow BODMAS order |
| Collect like terms | Core | Group terms with identical variable parts, combine coefficients |
| Expand single bracket | Core | $a(b+c) = ab + ac$, apply sign of coefficient to all terms |
| Expand two brackets | Core | Use FOIL rule, then collect like terms to simplify |
| Core factorisation | Core | Extract the highest common factor from all terms |
| Index laws (positive, zero, negative) | Core | $a^m×a^n=a^{m+n}$, $a^m÷a^n=a^{m-n}$, $(a^m)^n=a^{mn}$, $a^0=1$, $a^{-n}=1/a^n$ |
| Expand 3+ brackets | Extended | Expand first two brackets first, multiply result by remaining bracket |
| Special factorisation | Extended | Common factor, DOTS, grouping, $ax²+bx+c$, $a²x²−b²y²$ |
| Completing the square | Extended | Rewrite $ax²+bx+c$ as $a(x+p)² + q$ |
| Fractional indices | Extended | $a^{1/n}=ⁿ√a$, $a^{m/n}=(ⁿ√a)^m$ |
| Simple index equations | Core | Rewrite both sides with same base, equate exponents to solve |

## What's next

Now that you have mastered algebraic manipulation and indices for CIE IGCSE Maths 0580, you are ready to apply these foundational skills to more advanced algebra topics. These skills are prerequisites for solving linear and quadratic equations, rearranging formulae, and working with algebraic fractions, all of which appear frequently on both Core and Extended papers. You will also use index rules when working with standard form, sequences, and exponential graphs later in the syllabus. For Extended students, your understanding of completing the square will directly support solving quadratic equations and identifying key features of quadratic graphs in upcoming units. Be sure to practice past paper questions regularly to reinforce your speed and accuracy with these skills, as they are tested in almost every exam paper.

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