Algebra Toolkit (Edexcel IGCSE Maths): Revision notes

Section 1

What does index notation actually mean?

An index (or power/exponent) tells you how many times a base is multiplied by itself.

$a^n = \underbrace{a \times a \times \dots \times a}_{n \text{ times}}$

For example, $3^4 = 3 \times 3 \times 3 \times 3 = 81$ . In algebra the base can be a letter: $x^5 = x \times x \times x \times x \times x$ .

Special cases you must know:

$a^1 = a$
$a^0 = 1$ (for any $a \neq 0$ )
$a^{-n} = \dfrac{1}{a^n}$
$a^{\frac{1}{n}} = \sqrt[n]{a}$

Key termsbaseindex/exponentpower

Common mistake

$a^0 = 1$ , not $0$ — students often guess wrong here. Any non-zero number to the power 0 is always 1.

Example

$5^0 = 1$ , $x^0 = 1$ , but $0^0$ is undefined for IGCSE purposes.

Section 2

How do the index laws work?

These laws only apply when the bases are the same.

Multiplying: add the indices $a^m \times a^n = a^{m+n}$

Dividing: subtract the indices $a^m \div a^n = a^{m-n}$

Power of a power: multiply the indices $(a^m)^n = a^{mn}$

Negative indices: flip to a fraction $a^{-n} = \dfrac{1}{a^n}$

Fractional indices: roots $a^{\frac{1}{n}} = \sqrt[n]{a}, \qquad a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m$

Numbers with different bases (like $x^3 \times y^2$ ) cannot be combined using these laws.

Key termsindex lawsfractional indexnegative index

Exam tip

Say the law out loud as you use it: 'same base, multiplying, so I ADD the powers.' This stops you mixing up add/subtract/multiply.

Common mistake

$x^3 \times x^2 \neq x^6$ . Do NOT multiply the indices when multiplying terms — add them: $x^3 \times x^2 = x^5$ .

Example

$\dfrac{a^7}{a^3} = a^{7-3} = a^4$ ; $(x^2)^5 = x^{10}$ ; $8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2 = 4$ .

Section 3

What is substitution and how do you do it accurately?

Substitution means replacing letters in an expression with given numerical values and then evaluating.

Method:

Write out the expression.
Replace each letter with its value in brackets.
Apply BIDMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction) to simplify.

Example: find the value of $2x^2 - 3y$ when $x = -3$ and $y = 4$ . $2(-3)^2 - 3(4) = 2(9) - 12 = 18 - 12 = 6$

Using brackets around substituted values is essential, especially with negative numbers, so you don't lose a minus sign or misapply an index.

Key termssubstitutionBIDMAS

Common mistake

$(-3)^2 = 9$ , not $-9$ . Squaring a negative number always gives a positive result — the brackets matter.

Exam tip

Always substitute negative numbers in brackets: write $(-3)$ not just $-3$ , especially before an index.

Think of it like this

Think of substitution like a fill-in-the-blank template: the expression is the sentence structure, and you're just swapping in the given values without changing the structure.

Section 4

How do you collect like terms correctly?

Like terms have exactly the same letter(s) raised to exactly the same power(s). Only like terms can be added or subtracted.

$3x + 5x = 8x \qquad 7y^2 - 2y^2 = 5y^2$

But $3x$ and $5x^2$ are not like terms (different powers), and $3x$ and $3y$ are not like terms (different letters).

Method for simplifying an expression:

Identify groups of like terms.
Add/subtract the coefficients within each group, keeping the letter part unchanged.
Write the simplified expression, usually in descending order of powers.

Example: simplify $4x^2 + 3x - 2x^2 + 7 - 5x$ . $= (4x^2 - 2x^2) + (3x - 5x) + 7 = 2x^2 - 2x + 7$

Key termslike termscoefficientterm

Common mistake

$3x + 5x^2$ cannot be simplified to $8x^2$ or $8x^3$ — different powers of $x$ are different terms and stay separate.

Exam tip

Underline or colour-code matching terms before combining them, especially in longer expressions with several letters.

Must Know

$a^0 = 1$ for any non-zero $a$ ; $a^{-n} = \dfrac{1}{a^n}$
Multiplying same-base powers: ADD indices. Dividing: SUBTRACT indices. Power of a power: MULTIPLY indices.
$a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m$
Always substitute values in brackets, especially negatives, before applying BIDMAS
Only like terms (same letters, same powers) can be combined by adding/subtracting coefficients
Index laws only work when the bases are identical

That's the notes covered.

Carry on to the next subtopic.