When is not equal to 1?

The rule only applies when the base is not equal to 0, as is undefined in Edexcel IGCSE Mathematics A.

Use of Symbols

Edexcel International GCSE Mathematics A· 2.1· 12 min read

1. Algebraic Notation Conventions & Core Statement Types★☆☆☆☆⏱ 3 min

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Algebra uses letters (symbols) to represent unknown numbers or variables, following strict standard notation rules to avoid ambiguity. Always write numerical coefficients before variables, and omit multiplication signs between numbers and variables, or between multiple variables: write $3 x$ instead of $x 3$ or $3 \times x$ , and $2 ab$ instead of $2 \cdot a \cdot b$ .

📘 Definition

Core Algebraic Statement Types

Expression: A collection of algebraic terms with no equals sign, representing a single quantity.
Equation: A statement that two expressions are equal, true only for specific variable values.
Formula: A special type of equation that acts as a rule connecting two or more variables, used to calculate a quantity.
Identity: A statement that is true for all valid values of the variables involved, often written with the $\equiv$ symbol.

Example:

Classify each: (a) $v = u + a t$ (formula), (b) $4 x - 7 = 9$ (equation), (c) $5 p + 2 q - 3$ (expression), (d) $3 (p + q) \equiv 3 p + 3 q$ (identity)

📐 Worked Example

Rewrite the following using standard Edexcel IGCSE algebraic notation: (a) $5 \times a$ , (b) $b \times 4 \times c$ , (c) $y \times y \times y$

1
For (a): Omit the multiplication sign and write the number first: $5 a$
2
For (b): Write the number first, then letters in alphabetical order with no multiplication signs: $4 b c$
3
For (c): Use index notation for repeated multiplication of the same variable: $y^{3}$

✓ Quick check

Which of the following is an identity? A) $2 x + 5 = 11$ B) $x^{2} - 4 = 0$ C) $3 (x + 2) \equiv 3 x + 6$ D) $C = 2 π r$
Reveal answer
C —
A and B are equations (only true for specific x values), D is a formula for the circumference of a circle, and C is true for all values of x so it is an identity.

2. Integer Index Notation (Foundation & Higher)★★☆☆☆⏱ 3 min

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Index notation simplifies repeated multiplication of the same base. The general form is $a^{n}$ , where $a$ is the base and $n$ is the index (exponent). Rules apply for zero and negative integer indices, as well as positive ones.

📘 Definition

Integer Index Rules

Positive integer index: $a^{n} = a \times a \times ... \times a$ (n repetitions of the base)
Zero index: $a^{0} = 1$ for all $a \neq = 0$
Negative integer index: $a^{- n} = \frac{1}{a ^{n}}$ for all $a \neq = 0$

Example:

$x^{4} = x \times x \times x \times x$ , $y^{- 3} = \frac{1}{y ^{3}}$ , $z^{0} = 1$ (for $z \neq = 0$ )

📐 Worked Example

Rewrite the following using index notation, or without negative indices: (a) $k \times k \times k \times k \times k$ , (b) $\frac{1}{m ^{2}}$ , (c) $p^{0}$ (p≠0)

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For (a): 5 repetitions of k, so $k^{5}$
2
For (b): Apply the negative index rule: $m^{- 2}$
3
For (c): Apply the zero index rule: 1

3. Core Index Laws (Foundation & Higher)★★★☆☆⏱ 3 min

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The three core index laws only apply to expressions with the same base, and are used to simplify algebraic index expressions. You must recall these rules for exams, as they are not provided on the formula sheet.

Multiplication law: $x^{m} \times x^{n} = x^{m + n}$
Division law: $x^{m} \div x^{n} = x^{m - n}$
Power of a power law: $(x^{m})^{n} = x^{mn}$

📐 Worked Example

Simplify the following expressions, leaving answers with positive indices only: (a) $a^{2} \times a^{6}$ , (b) $b^{9} \div b^{4}$ , (c) $(c^{3})^{5}$ , (d) $d^{- 4} \times d^{6}$

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For (a): Apply the multiplication law, add indices: $2 + 6 = 8$ , so $a^{8}$
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For (b): Apply the division law, subtract indices: $9 - 4 = 5$ , so $b^{5}$
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For (c): Apply the power of a power law, multiply indices: $3 \times 5 = 15$ , so $c^{15}$
4
For (d): Add indices: $- 4 + 6 = 2$ , so $d^{2}$

4. Fractional Index Notation (Higher Tier Only)★★★★☆Higher only⏱ 2 min

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For Higher Tier, you need to use index notation involving fractional powers. The general rule for fractional indices is: $a^{\frac{m}{n}} = n a^{m} = (n a)^{m}$ , where the denominator of the fraction is the root, and the numerator is the power.

📐 Worked Example

Rewrite the following using root notation, or index notation as requested: (a) $x^{\frac{1}{2}}$ (as a root), (b) $y^{\frac{3}{4}}$ (as a root), (c) $5 z$ (as an index)

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For (a): Denominator 2 means square root, so $x$
2
For (b): Denominator 4 is fourth root, numerator 3 is cube, so $4 y^{3}$ or $(4 y)^{3}$
3
For (c): Fifth root is equivalent to power $\frac{1}{5}$ , so $z^{\frac{1}{5}}$

5. Common Pitfalls

Wrong move:

Writing $x 3$ instead of $3 x$ for 3 multiplied by x

Why:

Standard algebraic notation requires numerical coefficients to be written before variables, so $x 3$ is ambiguous and marked incorrect in exams.

Correct move:

Always write numerical coefficients first, e.g. $3 x$ , $7 ab$ .

Wrong move:

Applying index laws to expressions with different bases, e.g. simplifying $x^{2} \times y^{3}$ to $(x y)^{5}$

Why:

Index laws only apply when the bases of the terms are identical.

Correct move:

Only combine indices if the base is the same; $x^{2} \times y^{3}$ cannot be simplified further.

Wrong move:

Stating that $0^{0} = 1$

Why:

$0$ raised to the power of $0$ is undefined in Edexcel IGCSE mathematics, so the zero index rule only applies when the base is non-zero.

Correct move:

Always specify that $a^{0} = 1$ for $a \neq = 0$ .

Wrong move:

Subtracting indices when multiplying terms with the same base, e.g. $x^{3} \times x^{2} = x^{1}$

Why:

Multiplication of same-base terms requires adding indices, not subtracting.

Correct move:

Use the multiplication law: $x^{m} \times x^{n} = x^{m + n}$ , so $x^{3} \times x^{2} = x^{5}$ .

Wrong move:

Swapping the numerator and denominator of fractional indices, e.g. writing $a^{\frac{2}{3}}$ as $a^{3}$

Why:

The denominator of the fractional index is the root, and the numerator is the power, so this swaps the two values.

Correct move:

$a^{\frac{2}{3}} = 3 a^{2}$ or $(3 a)^{2}$ .

6. Quick Reference Cheatsheet

Concept	Rule	Example
Standard Notation	Write coefficients before variables, omit multiplication signs	$3 x$ , $5 ab$ , not $x 3$ , $5 \cdot a \cdot b$
Expression	No equals sign, collection of terms	$2 x + 7 y - 3$
Equation	True for specific variable values only	$3 x + 2 = 11$
Formula	Rule connecting multiple variables	$A = l w$ for area of rectangle
Identity	True for all valid variable values	$2 (x + 4) \equiv 2 x + 8$
Zero Index	$a^{0} = 1$ ( $a \neq = 0$ )	$(5 x)^{0} = 1$ ( $x \neq = 0$ )
Negative Index	$a^{- n} = 1/ a^{n}$ ( $a \neq = 0$ )	$x^{- 3} = 1/ x^{3}$
Multiplication Law	$x^{m} \times x^{n} = x^{m + n}$	$a^{4} \times a^{2} = a^{6}$
Division Law	$x^{m} \div x^{n} = x^{m - n}$	$b^{7} \div b^{3} = b^{4}$
Power of Power Law	$(x^{m})^{n} = x^{mn}$	$(c^{2})^{5} = c^{10}$
Fractional Index (Higher)	$a^{m / n} = n a^{m}$	$d^{2/3} = 3 d^{2}$

7. Frequently Asked

What is the difference between an equation and an identity?

An equation is only true for specific values of the variable, e.g. $2 x + 3 = 7$ only holds when $x = 2$ . An identity is true for all valid values of the variable, e.g. $2 (x + 1) \equiv 2 x + 2$ works for every possible value of $x$ .

When is $a^{0}$ not equal to 1?

The rule $a^{0} = 1$ only applies when the base $a$ is not equal to 0, as $0^{0}$ is undefined in Edexcel IGCSE Mathematics A.

Going deeper

official_specEdexcel 4MA1 2016 SpecificationRefer to section 2.1 for official topic guidance

What's Next

Now that you have mastered algebraic notation and core index laws, you are ready to apply these foundational skills to more advanced algebra topics in your Edexcel IGCSE Maths A course. The next step is to learn how to manipulate algebraic expressions, including expanding brackets and factorising, which build directly on the notation and index rules covered in this guide. You will also use these skills when solving linear and quadratic equations, rearranging formulae, and working with algebraic fractions later in the syllabus. For Higher Tier students, your understanding of fractional indices will be essential when working with surds and solving more complex algebraic problems in Papers 3H and 4H. Practice simplifying index expressions regularly to reinforce these rules, as they appear frequently across both Foundation and Higher exam papers.