Multiplying and Dividing using Indices
The first two index laws say that when you multiply powers of the same base you add the indices, and when you divide you subtract them: \(a^m \times a^n = a^{m+n}\) and \(a^m \div a^n = a^{m-n}\). This NSW Year 9 Core maths subtopic is Stage 5 Core content under NESA outcome MA5‑IND‑C‑01.
You will multiply and divide powers with numbers and pronumerals as bases, deal with the coefficients separately, as in \(18b^6 \div 3b^2 = 6b^4\), remember that \(m\) means \(m^1\), and handle negative coefficients and expressions with two or three different bases. You will also read a division written as a fraction, work a chain of multiplications and divisions, and spot the common mistake of multiplying the indices instead of adding them. It is part of Year 9 Core Indices and Scientific Notation.
Theory
The index laws make products and quotients of powers quick to simplify: multiplying powers of the same base adds the indices, and dividing them subtracts the indices. This Year 9 maths page shows why, then applies the laws to numbers, coefficients and terms with several pronumerals.
In \(a^5\), \(a\) is the base and \(5\) is the index (or power). It means five \(a\)s multiplied: \(a \times a \times a \times a \times a\). A pronumeral with no index written has index \(1\): \(a = a^1\).
When two powers of the same base are multiplied, the indices are added. When they are divided, the indices are subtracted.
The number in front of a term is its coefficient. Coefficients are multiplied or divided in the ordinary way; the index laws apply only to the bases.
Powers of different bases, such as \(p^2\) and \(q^3\), cannot be combined.
The two index laws on this page:
With coefficients:
| Expression | What happens | Answer |
|---|---|---|
| \(a^6 \times a^3\) | \(6 + 3\) | \(a^9\) |
| \(a^6 \div a^3\) | \(6 - 3\) | \(a^3\) |
| \(3a^6 \times 2a^3\) | \(3 \times 2\), then \(6 + 3\) | \(6a^9\) |
| \(a^6 \times b^3\) | bases differ | \(a^6b^3\) |
How to simplify with the index laws
- Write any missing index as \(1\).
- Work out the coefficients: multiply or divide them, keeping the sign.
- Combine each base on its own: add the indices to multiply, subtract to divide.
- Write the answer as the coefficient followed by each base in order. Leftover factors from the bottom stay on the bottom.
(i) Simplify \(5^4 \times 5^3\), leaving the answer in index form.
(ii) Simplify \(3^2 \times 3^2\) with an index law, then evaluate.
The bases match, so add the indices:
| \(5^4 \times 5^3\) | \(=\) | \(5^{4+3}\) |
| \(\) | \(=\) | \(\mathbf{5^7}\) |
| \(3^2 \times 3^2\) | \(=\) | \(3^{4}\) |
| \(\) | \(=\) | \(3 \times 3 \times 3 \times 3 = \mathbf{81}\) |
Check (ii) with the values: \(9 \times 9 = 81\). The base stays \(3\); it does not become \(9\).
Do the numbers, then each letter on its own. A letter with no index has index \(1\).
| \(-3 \times 5\) | \(=\) | \(-15\) |
| \(a^4 \times a^1\) | \(=\) | \(a^5\) |
| \(b^1 \times b^3\) | \(=\) | \(b^4\) |
| \(-3a^4b \times 5ab^3\) | \(=\) | \(\boldsymbol{-15a^5b^4}\) |
The coefficients multiply; only the indices add.
Divide the numbers, then subtract the indices:
| \(14 \div 4\) | \(=\) | \(\dfrac{14}{4} = \dfrac{7}{2}\) |
| \(m^8 \div m^3\) | \(=\) | \(m^{8-3} = m^5\) |
| \(14m^8 \div 4m^3\) | \(=\) | \(\boldsymbol{\dfrac{7m^5}{2}}\) |
When the numbers do not divide exactly, simplify them as a fraction.
A fraction is a division. Take it in three pieces:
| \(9 \div 3\) | \(=\) | \(3\) |
| \(\dfrac{p^6}{p^2}\) | \(=\) | \(p^4 \text{ (more on top)}\) |
| \(\dfrac{q^2}{q^5}\) | \(=\) | \(\dfrac{1}{q^3} \text{ (more underneath)}\) |
| \(\dfrac{9p^6q^2}{3p^2q^5}\) | \(=\) | \(\boldsymbol{\dfrac{3p^4}{q^3}}\) |
When the larger power is on the bottom, the leftover factors stay on the bottom.
Common pitfalls
Frequently asked questions
What happens to the indices when you multiply powers?
If the bases are the same, add the indices. For example \(a^3 \times a^4 = a^7\).
What happens to the indices when you divide powers?
If the bases are the same, subtract the indices. For example \(a^9 \div a^2 = a^7\).
What do you do with the numbers in front?
Multiply or divide the coefficients as ordinary numbers. The index laws only apply to the bases.
Can you add indices with different bases?
No. The index laws only work when the bases are the same. \(p^2 \times q^3\) stays as \(p^2q^3\).