The Index Laws
Master the index laws for Queensland Year 11 Mathematical Methods (QCAA). Indices, or powers, are a shorthand for repeated multiplication, and the index laws are the rules that let you multiply, divide and raise powers.
You will learn to apply the product, quotient and power laws, simplify powers of products and quotients, and handle the zero and negative index rules — algebra that prepares you for surds, exponentials and logarithms.
Every question with a fully worked solution.
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Theory
In Year 11 Mathematical Methods (QCAA, Unit 2), the index laws are the rules for multiplying, dividing and raising powers with the same base. This page covers the product, quotient and power laws, powers of products and quotients, and the zero and negative index rules, with fully worked simplifications.
A power such as \(a^{n}\) is written with a base \(a\) and an index (or exponent) \(n\); it means \(a\) multiplied by itself \(n\) times. The index laws are shortcuts that follow directly from this meaning.
The product law adds indices when powers of the same base are multiplied, and the quotient law subtracts them when they are divided. The power law multiplies indices when a power is raised to another power. A base to the zero index equals \(1\), and a negative index means a reciprocal, \(a^{-n}=\dfrac{1}{a^{n}}\).
For any base \(a\neq 0\) and integer indices \(m,\,n\):
How to simplify an index expression
- Numbers first: evaluate or simplify the numerical coefficients on their own.
- Group by base: collect the powers of each variable and apply one law at a time — add indices for a product, subtract for a quotient, multiply for a power of a power.
- Tidy up: write \(a^{0}=1\) and rewrite any negative index as a reciprocal so every index is a positive whole number.
Product law — multiply, so add the indices:
| \(a^{5}\times a^{3}\) | \(=\) | \(a^{5+3}\) |
| \(=\) | \(a^{8}\) |
Quotient law — divide, so subtract the index:
| \(a^{8}\div a^{2}\) | \(=\) | \(a^{8-2}\) |
| \(=\) | \(a^{6}\) |
\(a^{5}\times a^{3}\div a^{2}=a^{6}\).
Raise each factor to the power \(4\):
| \((2x^{3})^{4}\) | \(=\) | \(2^{4}\times (x^{3})^{4}\) |
Power law on \(x\) — multiply the indices:
| \(=\) | \(2^{4}\times x^{3\times 4}\) | |
| \(=\) | \(16\,x^{12}\) |
\((2x^{3})^{4}=16x^{12}\).
Numbers first:
| \(\dfrac{12}{4}\) | \(=\) | \(3\) |
Quotient law on each variable — subtract the indices:
| \(a\text{-part}\) | \(=\) | \(a^{5-2}=a^{3}\) |
| \(b\text{-part}\) | \(=\) | \(b^{-3-1}=b^{-4}\) |
Rewrite the negative index as a reciprocal:
| \(=\) | \(3a^{3}b^{-4}\) | |
| \(=\) | \(\dfrac{3a^{3}}{b^{4}}\) |
\(\dfrac{12a^{5}b^{-3}}{4a^{2}b}=\dfrac{3a^{3}}{b^{4}}\).
Expand the bracket with the power law:
| \((2x^{2}y)^{3}\) | \(=\) | \(2^{3}x^{6}y^{3}\) |
| \(=\) | \(8x^{6}y^{3}\) |
Multiply the numerator (product law on \(x\)):
| \(8x^{6}y^{3}\times x^{-4}\) | \(=\) | \(8x^{6+(-4)}y^{3}\) |
| \(=\) | \(8x^{2}y^{3}\) |
Divide by \(4xy^{2}\) (quotient law, and numbers):
| \(\dfrac{8x^{2}y^{3}}{4xy^{2}}\) | \(=\) | \(2x^{2-1}y^{3-2}\) |
| \(=\) | \(2xy\) |
\(\dfrac{(2x^{2}y)^{3}\times x^{-4}}{4xy^{2}}=2xy\).
Common pitfalls
Frequently asked questions
What are the index laws?
They are the rules for powers with the same base: \(a^{m}\times a^{n}=a^{m+n}\), \(\dfrac{a^{m}}{a^{n}}=a^{m-n}\), \((a^{m})^{n}=a^{mn}\), plus \(a^{0}=1\) and \(a^{-n}=\dfrac{1}{a^{n}}\).
When do you add and when do you multiply the indices?
Add the indices when you multiply powers of the same base; multiply the indices when a power is itself raised to a power, as in \((a^{m})^{n}=a^{mn}\).
What does a negative index mean?
A negative index means a reciprocal: \(a^{-n}=\dfrac{1}{a^{n}}\). For example \(2^{-3}=\dfrac{1}{8}\).
Why does anything to the power zero equal one?
Using the quotient law, \(a^{0}=\dfrac{a^{n}}{a^{n}}=1\) for any non-zero base \(a\).
Do the index laws work if the bases are different?
The product and quotient laws only apply when the bases are equal. With different bases, such as \(2^{3}\times 3^{2}\), you evaluate each power separately.