Indices, Standard Form & Surds

Q: Do I need to memorise index laws for the exam?

Yes, index laws are not provided on the 0580 formula sheet, so you must memorise all rules for Core and Extended content.

Q: When do I need to give answers as surds?

Extended tier questions will explicitly ask for exact answers, which require simplified surd form instead of rounded decimal values.

Mathematics· 1.7, 1.8, E1.18· 25 min read

1. 1. Indices (Core Tier Content)★★☆☆☆⏱ 8 min

Indices (or powers) are a shorthand for repeated multiplication of the same base number. For example, $3^{4} = 3 \times 3 \times 3 \times 3$ , where 3 is the base and 4 is the index.

📘 Definition

Core Index Laws

The following rules apply to all integer indices (positive, zero, negative):

$a^{m} \times a^{n} = a^{m + n}$
$a^{m} \div a^{n} = a^{m - n}$
$(a^{m})^{n} = a^{m \times n}$
$a^{0} = 1$ (for $a \neq = 0$ )
$a^{- n} = \frac{1}{a ^{n}}$

📐 Worked Example

Calculate the value of $\frac{2 ^{4} \times 2 ^{- 2}}{2 ^{1}}$

1
Apply the multiplication index law to the numerator:
$2^{4 + (-2)} = 2^2$
2
Apply the division index law to combine terms:
$2^{2 - 1} = 2^1 = 2$

✓ Quick check

What is the value of $5^{- 3}$ ?
- 15
- $\frac{1}{125}$
- -125
- -15
Reveal answer
$\frac{1}{125}$ —
Negative indices mean the reciprocal of the positive power: $5^{- 3} = \frac{1}{5 ^{3}} = \frac{1}{125}$

2. 2. Standard Form (Core + Extended Base Content)★★☆☆☆⏱ 7 min

Standard form is used to write extremely large or small numbers concisely, and is a common topic in both calculator and non-calculator papers.

📘 Definition

Standard Form Rule

A number is written in standard form as $A \times 1 0^{n}$ , where $1 \leq A < 10$ , and $n$ is an integer (positive for large numbers, negative for small numbers).

📐 Worked Example

(a) Convert 4 500 000 to standard form. (b) Convert $3.2 \times 1 0^{- 4}$ to ordinary form.

1
For (a): Move the decimal point 6 places left to get $A = 4.5$ , so $n = 6$ :
$4.5 \times 10^6$
2
For (b): Move the decimal point 4 places left to get the ordinary value:
$0.00032$

📐 Worked Example

Calculate $(2 \times 1 0^{3}) \times (3 \times 1 0^{2})$ , give your answer in standard form.

1
Multiply the $A$ values first:
$2 \times 3 = 6$
2
Multiply the powers of 10 using index laws:
$10^3 \times 10^2 = 10^5$
3
Combine terms, which is already in valid standard form:
$6 \times 10^5$

3. 3. Extended Tier: Fractional Indices★★★☆☆Extended only⏱ 5 min

Extended students are required to use fractional indices, which link power rules to root calculations.

📘 Definition

Fractional Index Rule

$a^{\frac{1}{n}} = n a$ , and $a^{\frac{m}{n}} = (n a)^{m} = n a^{m}$

📐 Worked Example

Calculate the value of $1 6^{\frac{3}{4}}$

1
Split the fractional index into root and power parts:
$16^{\frac{1}{4} \times 3}$
2
Calculate the 4th root of 16 first:
$16^{\frac{1}{4}} = 2$
3
Raise the result to the power of 3:
$2^3 = 8$

✓ Quick check

What is the value of $2 7^{\frac{2}{3}}$ ?
Reveal answer
9 —
$2 7^{\frac{1}{3}} = 3$ , so $3^{2} = 9$

4. 4. Extended Tier: Surds★★★★☆Extended only⏱ 7 min

Surds are irrational roots that cannot be simplified to whole numbers or fractions. You will be asked to manipulate surds to give exact answers in Extended papers.

📘 Definition

Surd Manipulation Rules

$a \times b = a \times b$
$\frac{a}{b} = \frac{a}{b}$
To rationalise a denominator of $\frac{1}{a}$ , multiply numerator and denominator by $a$

📐 Worked Example

Simplify $72$ fully.

1
Find the largest square factor of 72, which is 36:
$\sqrt{72} = \sqrt{36 \times 2}$
2
Apply the surd multiplication rule and simplify:
$\sqrt{36} \times \sqrt{2} = 6\sqrt{2}$

📐 Worked Example

Rationalise the denominator of $\frac{3}{2}$ .

1
Multiply numerator and denominator by $2$ to eliminate the surd in the denominator:
$\frac{3 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}}$
2
Simplify the denominator:
$\frac{3\sqrt{2}}{2}$

5. Common Pitfalls

Wrong move:

Calculating $a^{0} = 0$ for any base $a$

Why:

Confusing zero index with multiplication by zero

Correct move:

All non-zero numbers raised to the power of 0 equal 1

Wrong move:

Writing standard form with $A = 10$ , e.g. $10 \times 1 0^{4}$

Why:

Violates the rule $1 \leq A < 10$ for valid standard form

Correct move:

Rewrite as $1 \times 1 0^{5}$ to meet the standard form requirement

Wrong move:

Splitting $a + b = a + b$

Why:

Surd rules only apply to multiplication and division, not addition or subtraction

Correct move:

Simplify terms inside the root first if possible, or leave the expression unchanged

Wrong move:

Calculating $2^{- 3} = - 8$

Why:

Confusing negative index with a negative base value

Correct move:

Negative indices mean reciprocal: $2^{- 3} = \frac{1}{2 ^{3}} = \frac{1}{8}$

Wrong move:

Only multiplying the denominator by $a$ when rationalising

Why:

Changes the overall value of the fraction

Correct move:

Multiply both numerator and denominator by the same value to keep the fraction equivalent

6. Quick Reference Cheatsheet

Concept	Core Tier Rule	Extended Tier Addition
Indices	$a^{m} \times a^{n} = a^{m + n}$ , $a^{0} = 1$ , $a^{- n} = \frac{1}{a ^{n}}$	$a^{\frac{m}{n}} = (n a)^{m}$
Standard Form	$A \times 1 0^{n}$ ( $1 \leq A < 10$ ), convert to/from ordinary form, calculator arithmetic	No-calculator standard form arithmetic
Surds	Not assessed for Core	$ab = a \times b$ , simplify surds, rationalise denominators

7. Frequently Asked

Do I need to memorise index laws for the exam?

Yes, index laws are not provided on the 0580 formula sheet, so you must memorise all rules for Core and Extended content.

When do I need to give answers as surds?

Extended tier questions will explicitly ask for exact answers, which require simplified surd form instead of rounded decimal values.

Going deeper

syllabusCIE IGCSE Maths 0580 2025-2027 SyllabusOfficial specification for this topic

What's Next

You have now mastered all required content for indices, standard form, and surds for CIE IGCSE Maths 0580. These skills are foundational for nearly all other topics in the syllabus, including algebraic simplification, ratio and proportion, and geometry calculations. If you are taking Extended tier, make sure to practice surd rationalisation and non-calculator standard form questions regularly, as these are common high-mark question types in Paper 2. Next, you can move to ratio and percentage calculations, or apply your index skills to algebraic expressions.