What are indices in Mathematics, and why are they important in the IB MYP curriculum?

Indices, also known as exponents or powers, are a way of expressing repeated multiplication of a number by itself. For example, 3^4 means 3 multiplied by itself 4 times (3 x 3 x 3 x 3). In the IB MYP curriculum, understanding indices is crucial as they form the foundation for more advanced topics in Mathematics, such as algebra and calculus, and are essential for solving exponential equations and understanding scientific notation.

How do you simplify expressions with indices?

To simplify expressions with indices, you apply the laws of indices. These include the product of powers rule (a^m * a^n = a^(m+n)), the quotient of powers rule (a^m / a^n = a^(m-n)), and the power of a power rule ((a^m)^n = a^(m*n)). Understanding these rules helps in simplifying complex expressions and solving equations efficiently in the IB MYP Mathematics curriculum.

What is the difference between positive and negative indices?

Positive indices indicate the number of times a base number is multiplied by itself, while negative indices represent the reciprocal of the base raised to the corresponding positive index. For example, a^-n = 1/a^n. This concept is important in the IB MYP Mathematics curriculum as it helps students understand the behavior of numbers in different contexts, such as growth and decay in real-world applications.

How do fractional indices work, and what do they represent?

Fractional indices represent roots of numbers. For example, a^(1/n) is the nth root of a, and a^(m/n) is the nth root of a raised to the power of m. Understanding fractional indices is essential in the IB MYP Mathematics curriculum as it allows students to solve equations involving roots and powers, which are common in various mathematical and scientific contexts.

Can you explain the zero index rule and its significance?

The zero index rule states that any non-zero number raised to the power of zero is equal to one (a^0 = 1). This rule is significant because it helps maintain the consistency of the laws of indices and is a fundamental concept in the IB MYP Mathematics curriculum. It is particularly useful in simplifying expressions and solving equations where terms may cancel out, leaving a base raised to the power of zero.

Why is "Writing $2^3 \times 5^4 = 10^7$." a common mistake in Indices?

Why it happens: Trying to apply the multiplication law with different bases. How to avoid it: Index laws only combine SAME-BASE powers. 2^3 × 5^4 has no simplification — compute as 8 × 625 = 5000.

Why is "Writing $5^0 = 0$." a common mistake in Indices?

Why it happens: Confusing a^0 with a × 0. How to avoid it: a^0 = 1 for any a 0. Comes from a^n / a^n = a^0 = 1.

Why is "Treating $3^{-2}$ as $-9$." a common mistake in Indices?

Why it happens: Mixing negative sign with negative index. How to avoid it: 3^-2 = 1/3^2 = 1/9 — it's a SMALL positive number, not a negative.

NumberIB MYP Maths Extended Level

Indices

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Indices — IB MYP Mathematics (Extended): laws of exponents and roots

Powers compress repeated multiplication. This MYP Mathematics Extended note covers the five core index laws, fractional and negative indices, and how to manipulate them confidently.

What you’ll learn

Mapped to the IB MYP Mathematics subject guide (2026 onwards).

MYP Mathematics A — Apply the index laws confidently.
MYP Mathematics A — Evaluate fractional and negative indices.
MYP Mathematics C — Show clear index manipulation in algebraic working.
MYP Mathematics D — Use indices in scientific contexts (growth, scale).

The five index laws

Memorise them — they're used everywhere in algebra.

Every index law follows from the basic definition $a^n = \underbrace{a \times a \times \ldots \times a}_{n \text{ times}}$ .

Law	Rule	Example
Multiplication	$a^m \times a^n = a^{m+n}$	$2^3 \times 2^4 = 2^7 = 128$
Division	$a^m \div a^n = a^{m-n}$	$5^7 \div 5^4 = 5^3 = 125$
Power of a power	$(a^m)^n = a^{mn}$	$(3^2)^4 = 3^8$
Power of a product	$(ab)^n = a^n b^n$	$(2x)^3 = 8x^3$
Power of a quotient	$\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$	$\left(\dfrac{3}{4}\right)^2 = \dfrac{9}{16}$

The base is the number being multiplied; the index (or exponent) says how many times. So $2^3 = 2 \times 2 \times 2 = 8$.

The first three laws are the most important — and they only work when the BASES MATCH. You cannot simplify $2^3 \times 3^4$ using the multiplication rule.

$a^m \times a^n = a^{m+n}$ (ADD when multiplying same-base powers).
$a^m \div a^n = a^{m-n}$ (SUBTRACT when dividing).
$(a^m)^n = a^{mn}$ (MULTIPLY when raising to another power).

Special powers: 0, negative and fractional

These follow from the index laws.

Power of zero: $a^0 = 1$ for any $a \ne 0$ .

Why? Using the division rule: $a^n \div a^n = a^{n-n} = a^0$ . But anything divided by itself is 1. So $a^0 = 1$ .

Negative indices: $a^{-n} = \dfrac{1}{a^n}$ .

Why? Using the division rule again: $a^0 \div a^n = a^{0-n} = a^{-n}$ . And $a^0 \div a^n = \dfrac{1}{a^n}$ . So $a^{-n} = \dfrac{1}{a^n}$ .

Examples:

$3^{-2} = \dfrac{1}{3^2} = \dfrac{1}{9}$ .
$\left(\dfrac{2}{5}\right)^{-1} = \dfrac{5}{2}$ (negative power flips the fraction).
$\left(\dfrac{1}{4}\right)^{-2} = 4^2 = 16$ .

Fractional indices: $a^{1/n} = \sqrt[n]{a}$ .

Why? Using the power-of-a-power rule: $\left(a^{1/n}\right)^n = a^{(1/n)\cdot n} = a^1 = a$ . So $a^{1/n}$ is the number that gives $a$ when raised to the power $n$ — which is exactly the $n$ th root.

Examples:

$9^{1/2} = \sqrt{9} = 3$ .
$27^{1/3} = \sqrt[3]{27} = 3$ .
$16^{3/4} = (16^{1/4})^3 = 2^3 = 8$ .

General rule: $a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$ .

$a^0 = 1$ (for $a \ne 0$ ).
$a^{-n} = \dfrac{1}{a^n}$ .
$a^{1/n} = \sqrt[n]{a}$ , so $a^{m/n} = (\sqrt[n]{a})^m$ .
Negative power of a fraction = flip and use positive power.

Strategy for combined expressions

Use the laws step by step.

Many MYP Extended questions combine several index laws. Work methodically:

Apply BRACKETS and any power-of-a-power steps.
Apply MULTIPLICATION and DIVISION rules to combine same-base powers.
Convert NEGATIVE indices to reciprocals if you need a numerical answer.
Convert FRACTIONAL indices to roots if needed.

Worked example. Simplify $\dfrac{(2^3)^2 \times 2^{-4}}{2^{-1}}$ .

Power of a power: $(2^3)^2 = 2^6$ . Expression becomes $\dfrac{2^6 \times 2^{-4}}{2^{-1}}$ .
Top: $2^6 \times 2^{-4} = 2^{6-4} = 2^2$ . Expression: $\dfrac{2^2}{2^{-1}}$ .
Division: $2^{2 - (-1)} = 2^3 = 8$ .

Worked example (Extended). Simplify $(8x^6 y^{-3})^{2/3}$ .

Power of a product: $8^{2/3} \cdot (x^6)^{2/3} \cdot (y^{-3})^{2/3}$ .
$8^{2/3} = (8^{1/3})^2 = 2^2 = 4$ .
$(x^6)^{2/3} = x^4$ .
$(y^{-3})^{2/3} = y^{-2} = \dfrac{1}{y^2}$ .
Combine: $\dfrac{4 x^4}{y^2}$ .

This kind of question is bread-and-butter for Extended Level — and a great place to lose easy marks if you skip steps.

Apply power-of-a-power first to clear brackets.
Combine same-base powers using multiplication/division rules.
Negative or fractional powers can stay as indices OR be converted — depending on what the question wants.

How it’s examined

Criterion A: laws of indices on numerical and algebraic expressions. Criterion C: clear step-by-step index manipulation.

Sources: IB MYP Mathematics subject guide (IBO, official). Last reviewed 2026-05-25.

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Step-by-step worked examples — Indices

Step-by-step solutions to past-paper-style questions on indices, written exactly the way a tutor would explain them at the board.

1Combine indices using the laws
Getting started• index laws
Question
Simplify $3^5 \times 3^4 \div 3^2$ .
Step-by-step solution
1. Step 1
  Multiplication rule: $3^5 \times 3^4 = 3^{5+4} = 3^9$ .
2. Step 2
  Division rule: $3^9 \div 3^2 = 3^{9-2} = 3^7$ .
3. Step 3
  If asked for a number: $3^7 = 2187$ .
Answer
$3^7$ (or $2187$ ).
2Negative index
Building confidence• negative
Question
Evaluate $2^{-3} + \left(\dfrac{1}{2}\right)^{-2}$ .
Step-by-step solution
1. Step 1
  $2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}$ .
2. Step 2
  $\left(\dfrac{1}{2}\right)^{-2} = 2^2 = 4$ (negative power flips the fraction).
3. Step 3
  Sum: $\dfrac{1}{8} + 4 = \dfrac{1}{8} + \dfrac{32}{8} = \dfrac{33}{8}$ .
Answer
$\dfrac{33}{8}$
3Fractional index
Building confidence• fractional
Question
Evaluate $64^{2/3}$ .
Step-by-step solution
1. Step 1
  Split: $64^{2/3} = (64^{1/3})^2$ .
2. Step 2
  Cube root: $64^{1/3} = 4$ (because $4^3 = 64$ ).
3. Step 3
  Square: $4^2 = 16$ .
Answer
$16$
4Algebraic indices
Stretch• algebra
Question
Simplify $\dfrac{6 a^4 b^{-2}}{2 a^{-1} b^3}$ .
Step-by-step solution
1. Step 1
  Numerical: $\dfrac{6}{2} = 3$ .
2. Step 2
  $a^4 \div a^{-1} = a^{4 - (-1)} = a^5$ .
3. Step 3
  $b^{-2} \div b^3 = b^{-2-3} = b^{-5}$ .
4. Step 4
  Combine: $3 a^5 b^{-5} = \dfrac{3 a^5}{b^5}$ .
Answer
$\dfrac{3 a^5}{b^5}$

Key Definitions and Keywords — Indices

Definitions to memorise and the exact keywords mark schemes credit for indices answers — sharpened from recent examiner reports for the 2026 IB MYP Mathematics (Extended) sitting.

Base
Examiner keyword
The number being multiplied in a power. In $a^n$ , $a$ is the base.
Index (exponent)
Examiner keyword
The number that says how many times the base is multiplied. In $a^n$ , $n$ is the index.
$n$ th root
$\sqrt[n]{a}$ is the number that, raised to the $n$ th power, gives $a$ . Equal to $a^{1/n}$ .

Common Mistakes and Misconceptions — Indices

The traps other students keep falling into on indices questions — taken from recent IB MYP Mathematics (Extended) examiner reports and mark schemes — and how to avoid them.

✕Writing $2^3 \times 5^4 = 10^7$ .
Why it happens
Trying to apply the multiplication law with different bases.
How to avoid it
Index laws only combine SAME-BASE powers. $2^3 \times 5^4$ has no simplification — compute as $8 \times 625 = 5000$ .
✕Writing $5^0 = 0$ .
Why it happens
Confusing $a^0$ with $a \times 0$ .
How to avoid it
$a^0 = 1$ for any $a \ne 0$ . Comes from $a^n / a^n = a^0 = 1$ .
✕Treating $3^{-2}$ as $-9$ .
Why it happens
Mixing negative sign with negative index.
How to avoid it
$3^{-2} = \dfrac{1}{3^2} = \dfrac{1}{9}$ — it's a SMALL positive number, not a negative.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

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Indices — frequently asked questions

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Indices

Short Study Notes in the form of Slides

Short Study Notes — Indices

Detailed Notes

What you’ll learn

How it’s examined

1Combine indices using the laws

2Negative index

3Fractional index

4Algebraic indices

Base

Index (exponent)

nnnth root

✕Writing 23×54=1072^3 \times 5^4 = 10^723×54=107.

✕Writing 50=05^0 = 050=0.

✕Treating 3−23^{-2}3−2 as −9-9−9.

Practice questions

Past paper style quiz

Assess your understanding

Indices — frequently asked questions

$n$ th root

✕Writing $2^3 \times 5^4 = 10^7$ .

✕Writing $5^0 = 0$ .

✕Treating $3^{-2}$ as $-9$ .