---
title: "Index laws Practice Questions & Worked Answers"
description: "Multiply, divide and raise powers without expanding them, including zero and negative indices. Free Years 8 to 9 practice with worked solutions for ever..."
image: "https://nobleeducators.com/og-image.jpg"
url: "https://nobleeducators.com/australia/practice/index-laws"
---

Your Australian school pathway · Noble Educators

# Index laws practice questions and worked solutions

Index laws let you multiply, divide and raise powers without ever expanding them. They look like a short list of rules to memorise, which is exactly why they get confused with each other under pressure. Every one of them follows from what an index means, and a student who can rebuild a law from that meaning will never mix up adding indices with multiplying them.

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## Why the laws work at all

An index is shorthand for repeated multiplication, so x⁵ means five copies of x multiplied together. Multiplying x⁵ by x³ therefore puts five copies alongside three copies, giving eight in total — which is why multiplying powers of the same base adds the indices rather than multiplying them.

Division cancels rather than accumulates. In x⁷ divided by x², two copies on top cancel with the two underneath and five remain, so the indices subtract. Rebuilding either law by writing out the copies takes fifteen seconds and settles any doubt about which operation applies.

The same-base condition is not decoration. There is no index law that simplifies x³ multiplied by y³ into a single power, because the copies are of different things and nothing cancels or combines. Questions mixing bases are testing whether students apply the laws mechanically or with understanding.

## The law students most often confuse

Raising a power to a power multiplies the indices, so (x³)⁴ is x¹². This is the law most frequently swapped with the multiplication law, because both involve two indices and a single base. The bracket is the reliable signal: brackets multiply the indices, a multiplication sign between separate terms adds them.

The confusion compounds when a coefficient is present. In (2x³)⁴ the four applies to everything inside the bracket, giving 16x¹², and students who apply it only to the x lose the coefficient entirely. Writing the bracket out as four copies once is usually enough to fix the habit permanently.

## Zero and negative indices

Anything non-zero raised to the power of zero equals one, and this follows directly from the division law: x³ divided by x³ is x⁰ by the rule, and any non-zero quantity divided by itself is one. Presenting it as an arbitrary fact to memorise is why students doubt it under pressure.

A negative index means a reciprocal, not a negative value. So 2⁻³ is one over 2³, which is one eighth — a positive number. Students who answer negative eight have negated the value instead of inverting it, and this single misreading accounts for a large share of lost marks in the topic.

Combining the two produces expressions like 3x⁻² where only the x carries the negative index, so the result is three over x squared rather than one over three x squared. Bracketing carefully when you rewrite is what keeps the coefficient in the right place.

## Fractional indices and surds

A fractional index is a root: x to the power of one half is the square root of x, and x to the power of two thirds is the cube root of x squared. This connects two topics that are often taught separately, and recognising the link means surd questions can be answered with index laws instead of a second set of rules.

Senior courses lean on this heavily, particularly in calculus where a term such as one over the square root of x must be rewritten as x to the power of negative one half before it can be differentiated. Students who are fluent with fractional indices find that transition considerably easier than those who are not.

## Practising the laws so they stay learned

Index laws are unusually vulnerable to being forgotten between the unit and the exam, because they are learned as a list and a list decays quickly. Mixed sets, where consecutive questions require different laws, hold up far better than blocked practice on one law at a time even though they feel harder while you are doing them.

The other habit worth building is checking a doubtful result with small numbers. If you are unsure whether (x³)⁴ is x⁷ or x¹², substitute two for x and evaluate both sides: eight to the fourth is 4096, and two to the twelfth is also 4096, which settles it in seconds without needing to recall the rule.

## Practise it now

5 questions

Choose an answer and the method appears straight away. Nothing is recorded and there is no time limit — work through them at whatever pace is useful, and read the working even on the ones you get right.

1. Question 1. Simplify x⁵ × x³.x⁸x¹⁵x²2x⁸
2. Question 2. Simplify x⁷ ÷ x².x⁵x⁹x³·⁵x²
3. Question 3. Simplify (x³)⁴.x¹²x⁷x⁸¹4x³
4. Question 4. What is the value of x⁰, where x is any non-zero number?10xUndefined
5. Question 5. Write 2⁻³ as a fraction.1/8−8−61/6

Years 8 to 9 · Australian Curriculum v9.0 content descriptors AC9M8N02, AC9M9N01. Descriptor codes are published by ACARA and are quoted so you can check this practice against your school’s scope and sequence. Noble Educators is an independent tutoring provider and is not affiliated with ACARA.

Common questions

## What families ask about this

### What year are index laws taught in Australia?

They are introduced in Year 8 with positive integer indices and extended in Year 9 to include zero and negative indices, with fractional indices and scientific notation following in Year 10 under the Australian Curriculum.

### Why is anything to the power of zero equal to one?

Because dividing a power by itself gives one, and the division law says the same expression equals the base to the power of zero. Both statements describe the same quantity, so the base to the power of zero must be one for any non-zero base.

### Do index laws work when the bases are different?

Not for combining them into a single power. You can simplify each base separately, and there are laws for products and quotients raised to a power, but there is no rule that turns two different bases with the same index into one term.

### How do index laws relate to scientific notation?

Scientific notation is index laws applied to powers of ten, so multiplying two numbers in that form means multiplying the decimal parts and adding the exponents. Students comfortable with the laws generally find scientific notation straightforward rather than a separate topic.

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| Option | Typical cost per session | Format |
| --- | --- | --- |
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| Tutoring centre | $45–$90 | Small group, sometimes 1-on-1 |
| Online marketplace tutor | $30–$70 | 1-on-1, live |

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