Index Laws Explained: All 7 Laws of Indices with Examples and Practice (Years 8 to 10)

Index laws: all 7 laws of indices, explained with examples

Index laws are shortcuts for simplifying powers. There are seven to learn, and they all come from what a power means. Here’s each law with an example, the mistakes that cost marks, and practice questions with answers.

By Ryan, founder of Bayside Academics, 15 years in private tutoring · Updated 6 November 2026

Quick answer

With the same base: when multiplying, add the indices (aᵐ × aⁿ = aᵐ⁺ⁿ); when dividing, subtract them (aᵐ ÷ aⁿ = aᵐ⁻ⁿ); for a power of a power, multiply them ((aᵐ)ⁿ = aᵐⁿ). Anything (except 0) to the power 0 is 1, and a negative index means one over: a⁻ⁿ = 1/aⁿ.

What an index means

In $2^5$ , the base is 2 and the index (or exponent, or power) is 5. It means multiply 2 by itself 5 times:

2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32

Every index law comes from this. If you forget one in a test, write the powers out in full and you'll see the pattern.

The 7 index laws

Laws 1 to 3 only work when the bases are the same.

1. Multiplying: add the indices

a^m \times a^n = a^{m+n}

Example: $x^4 \times x^3 = x^7 \qquad 2^3 \times 2^5 = 2^8 = 256$

2. Dividing: subtract the indices

a^m \div a^n = a^{m-n}

Example: $y^9 \div y^4 = y^5 \qquad \frac{5^6}{5^4} = 5^2 = 25$

3. Power of a power: multiply the indices

(a^m)^n = a^{mn}

Example: $(x^3)^4 = x^{12}$

4. Power of a product

(ab)^n = a^n b^n

Example: $(2x)^3 = 2^3x^3 = 8x^3$

5. Power of a quotient

\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}

Example: $\left(\frac{x}{3}\right)^2 = \frac{x^2}{9}$

6. The zero index

a^0 = 1 \quad (a \neq 0)

Example: $5^0 = 1 \qquad (3x)^0 = 1 \qquad 3x^0 = 3$

7. Negative indices

a^{-n} = \frac{1}{a^n}

Example: $2^{-3} = \frac{1}{8} \qquad x^{-2} = \frac{1}{x^2}$

Combining the laws

Deal with the numbers (coefficients) and each letter separately, one law at a time.

Worked example

Simplify $(3x^2y^3)^2 \times 2xy^{-1}$ .

Power of a product, then power of a power:

(3x^2y^3)^2 = 9x^4y^6

Multiply: numbers $9 \times 2 = 18$ , add the indices on each letter:

9x^4y^6 \times 2xy^{-1} = 18x^{5}y^{5}

Worked example

Simplify $\dfrac{12a^5b^2}{4a^2b^5}$ and write the answer with positive indices.

\frac{12}{4} = 3, \quad a^{5-2} = a^3, \quad b^{2-5} = b^{-3}

3a^3b^{-3} = \frac{3a^3}{b^3}

Fractional indices (Year 10A)

A fractional index means a root: the bottom number is the root, the top is the power.

a^{\frac{1}{n}} = \sqrt[n]{a}

a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m

For example, $8^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$ , and $16^{\frac{3}{4}} = \left(\sqrt[4]{16}\right)^3 = 2^3 = 8$ . Take the root first so the numbers stay small.

Common mistakes that cost marks

Adding the coefficients. 2x³ × 3x⁴ = 6x⁷: multiply the numbers, add the indices.
Multiplying indices when multiplying powers. x³ × x⁴ is x⁷, not x¹².
Using the laws on addition. x³ + x² can’t be simplified to x⁵.
The zero index on the wrong part. 3x⁰ = 3 × 1 = 3, but (3x)⁰ = 1.
Thinking a negative index makes it negative. 2⁻³ is 1/8, a positive number.
Forgetting to raise the number. (2x)³ = 8x³, not 2x³.

Practice questions with answers

Try each one before you open the answer.

Simplify $a^5 \times a^6$
Show the answer
$a^{11}$
Simplify $4x^3 \times 5x^2$
Show the answer
Multiply the numbers and add the indices: $20x^5$
Simplify $m^{10} \div m^4$
Show the answer
$m^6$
Simplify $18y^7 \div 6y^2$
Show the answer
Divide the numbers and subtract the indices: $3y^5$
Simplify $(p^4)^3$
Show the answer
$p^{12}$
Simplify $(2x^3)^4$
Show the answer
$2^4 x^{12} = 16x^{12}$
Evaluate $7^0 + 4x^0$
Show the answer
$1 + 4 \times 1 = 5$
Write $5^{-2}$ as a fraction
Show the answer
$\dfrac{1}{5^2} = \dfrac{1}{25}$
Simplify $(3a^2b)^2 \div a^3b^4$ , with positive indices
Show the answer
$9a^4b^2 \div a^3b^4 = 9a^{1}b^{-2} = \dfrac{9a}{b^2}$
(Year 10A) Evaluate $27^{\frac{2}{3}}$
Show the answer
$\left(\sqrt[3]{27}\right)^2 = 3^2 = 9$

Want more? Each year level below has a free printable PDF of 10 questions with answers, and a full 20-question test.

Index laws by year level

Following the Victorian Curriculum 2.0. Schools sequence topics differently, so check your course.

Year 8

Exponents

Index notation and the first index laws with whole-number indices.

Worksheets and topic test

Year 9

Exponent laws

All the index laws with variables, including the zero and negative indices.

Worksheets and topic test · Free PDF with answers

Year 9

Scientific notation

Using powers of 10 to write very large and very small numbers.

Worksheets and topic test · Free PDF with answers

Year 10

Index laws with algebra

Index laws in algebra: simplifying and expanding expressions.

Worksheets and topic test · Free PDF with answers

Year 10A

Surds and fractional indices

Fractional indices and surds.

Worksheets and topic test · Free PDF with answers

Index laws: common questions

What are the 7 index laws?

Multiplying (add the indices), dividing (subtract them), power of a power (multiply them), power of a product, power of a quotient, the zero index (anything to the power 0 is 1) and negative indices (a to the power −n is 1 over a to the power n).

Are indices the same as exponents and powers?

Yes. Index, exponent and power all mean the small raised number. Australian schools usually say index (plural indices) or exponent.

Do the index laws work when adding?

No. The laws are for multiplying and dividing powers with the same base. x² + x³ can't be simplified to a single power.

Why is anything to the power of zero equal to 1?

Using the dividing law, a³ ÷ a³ = a⁰. But anything divided by itself is 1, so a⁰ = 1 (as long as a isn't 0).

What year level learns index laws?

In the Victorian Curriculum 2.0, index laws start in Year 8 with whole-number indices, extend in Year 9 to negative indices and scientific notation, and are used in Year 10 algebra. Year 10A adds fractional indices and surds.

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What an index means

The 7 index laws

Combining the laws

Fractional indices (Year 10A)

Common mistakes that cost marks

Practice questions with answers

Index laws by year level

Index laws: common questions

More maths help, Years 7 to 10

Got an index laws question wrong in homework?