Summary of Index Laws
Summary of Index Laws brings the multiplication, division, power and zero index laws together in single expressions. In NSW Year 9 Advanced maths the skill is applying them in a sensible order: brackets first, then multiplying and dividing (NESA outcome MA5‑IND‑C‑01).
You will simplify expressions such as \(\frac{(2x^3)^2}{4x}\) and \(\frac{10k^7}{(2k)^3}\), evaluate numerical powers using an index law, and pick the correct working from four students. Longer questions combine several brackets in a fraction, evaluate the result for given values, find unknown indices \(p\) and \(q\), locate the first mistake in a student's working, and simplify powers with \(n\) in the index. It is part of Year 9 Advanced Indices and Scientific Notation.
Every question with a fully worked solution.
- Summary of Index Laws - Video - Laws of Indices Watch
Theory
The index laws say what happens to the indices when powers of the same base are multiplied, divided or raised to a power. This Year 9 Advanced maths page puts all five laws together with a four-step plan for mixed expressions, substitution, finding mistakes in working and solving for unknown indices.
The index laws apply to powers of the same base. Each one comes from writing the powers out as repeated factors.
Multiplication joins factors (add the indices). Division cancels factors (subtract the indices). A power of a power repeats a group of factors (multiply the indices). A power of a product raises every factor in the bracket. The zero index gives \(a^0 = 1\).
A mixed question uses several laws. Work in a fixed order: brackets first, because they change the coefficients and indices you collect afterwards.
The laws do not cover addition or subtraction. \(x^3 + x^4\) has unlike terms and cannot be simplified, and \(2x^3 + 5x^3 = 7x^3\) is collecting like terms, not an index law.
For non-zero bases \(a\) and \(b\):
| Law | Rule | Example |
|---|---|---|
| Multiplication | \(a^m \times a^n = a^{m+n}\) | \(w^6 \times w^2 = w^8\) |
| Division | \(a^m \div a^n = a^{m-n}\) | \(w^6 \div w^2 = w^4\) |
| Power of a power | \((a^m)^n = a^{mn}\) | \((w^6)^2 = w^{12}\) |
| Power of a product | \((ab)^n = a^nb^n\) | \((3w)^2 = 9w^2\) |
| Power of a quotient | \(\left(\dfrac{a}{b}\right)^n\) \(= \dfrac{a^n}{b^n}\) | \(\left(\dfrac{w}{3}\right)^2\) \(= \dfrac{w^2}{9}\) |
| Zero index | \(a^0 = 1\) | \(6w^0 = 6\) |
The four-step plan
- Expand brackets: raise every factor inside, signs included.
- Collect the top: multiply the coefficients and add the indices of each pronumeral.
- Collect the bottom the same way.
- Divide, base by base: divide the coefficients (leave a fraction if needed) and subtract the indices. A base with the larger index underneath stays underneath.
- Check: substitute \(x = 1\) or \(x = 2\) into the question and your answer.
Other question types
- Written \(\div\) signs: \(20k^7 \div (4k^2) \times 3k\) is worked left to right: \(5k^5 \times 3k = 15k^6\).
- Numerical bases: the same laws work on numbers: \((7^2)^3 \times 7 \div 7^4 = 7^{3}\). A bracketed negative base keeps its sign by the odd/even rule.
- Which statement is correct? Test each with a number, such as \(x = 2\). A correct simplification gives equal values.
- Find the mistake: redo each line yourself and compare; the first line that disagrees holds the mistake (Example 3).
- One index only: if a question asks for just the index of \(n\), follow only the powers of \(n\) through the laws.
Expand the brackets, collect the top and bottom, then divide.
| \((3m^2n)^3\) | \(=\) | \(27m^6n^3\) |
| \((6m^3n^2)^2\) | \(=\) | \(36m^6n^4\) |
| \(27m^6n^3 \times 2mn^4\) | \(=\) | \(54m^7n^7\) |
| \(\dfrac{54m^7n^7}{36m^6n^4}\) | \(=\) | \(\dfrac{54}{36}m^{7-6}n^{7-4}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{3mn^3}{2}}\) |
\(\dfrac{54}{36} = \dfrac{3}{2}\) does not simplify to a whole number, so it stays as a fraction.
Simplify first: the numbers are far smaller afterwards.
| \((-2xy^3)^3\) | \(=\) | \(-8x^3y^9\) |
| \((3x^2y)^2\) | \(=\) | \(9x^4y^2\) |
| \(\text{top}\) | \(=\) | \(-72x^7y^{11}\) |
| \(\dfrac{-72x^7y^{11}}{24x^5y^9}\) | \(=\) | \(\boldsymbol{-3x^2y^2}\) |
Substitute, keeping the negative value in brackets:
| \(-3x^2y^2\) | \(=\) | \(-3 \times (-2)^2 \times 3^2\) |
| \(\) | \(=\) | \(-3 \times 4 \times 9\) |
| \(\) | \(=\) | \(\mathbf{-108}\) |
\((-2)^2 = 4\) is positive, so the sign of the answer comes only from the \(-3\).
Check each line against the laws.
| \((2x^2)^3\) | \(=\) | \(2^3x^{6} = 8x^6\) |
Noah wrote \(6x^6\): he multiplied \(2 \times 3\) instead of working out \(2^3\). The mistake is in Line 1. The index of \(x\) was right.
| \(\dfrac{12x^9}{8x^6}\) | \(=\) | \(\dfrac{12}{8}x^{9-6}\) |
| \(\) | \(=\) | \(\boldsymbol{\dfrac{3x^3}{2}}\) |
Check with \(x = 1\): \(\dfrac{12}{8} = 1.5\), and \(\dfrac{3}{2} = 1.5\). Noah's \(2x^3\) gives \(2\).
Simplify the left side in terms of \(p\) and \(q\), then match it with the right side.
| \((2x^3y^p)^4\) | \(=\) | \(16x^{12}y^{4p}\) |
| \(\dfrac{16x^{12}y^{4p}}{8x^{q}y^5}\) | \(=\) | \(2x^{12-q}y^{4p-5}\) |
Match the indices of \(x\), then of \(y\):
| \(12 - q\) | \(=\) | \(5 \ \Rightarrow\ q = \mathbf{7}\) |
| \(4p - 5\) | \(=\) | \(19 \ \Rightarrow\ p = \mathbf{6}\) |
The coefficients also match (\(16 \div 8 = 2\)), which confirms the setting out.
Common pitfalls
Frequently asked questions
What are the index laws?
Multiply: add the indices. Divide: subtract the indices. Power of a power: multiply the indices. Power of a product: raise every factor. Zero index: \(a^0 = 1\).
What order do you use the index laws in?
Expand any brackets first, then collect the top, then collect the bottom, and finally divide base by base. Brackets come first because they change the coefficients and indices.
Can index laws be used when adding terms?
No. The index laws are for multiplying, dividing and raising to powers. Added terms combine only if they are like terms, such as \(2x^3 + 5x^3 = 7x^3\).
How can I check my index simplification?
Substitute a small number such as \(2\) for the pronumeral in the original expression and in your answer. If the two values differ, there is a mistake.
How do you find an unknown index?
Simplify the expression in terms of the unknown, then match it with the given answer: the index of each pronumeral must be the same on both sides. Solve the equations that result.