# Multiplying and Dividing Terms

> Multiply numbers and letters together, use index notation, and cancel when you divide.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/multiplying-and-dividing-terms/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: Tiles on the patio
Grandpa is tiling a patio. Each tile is $a$ centimetres long. He lays 5 rows, and each row has 4 tiles.

Zara counts the tiles: $5 \times 4 = 20$ tiles. Each one is $a$ long, so if they were laid end to end the line would be $20a$ centimetres long.

"So $5 \times 4a$ is $20a$," she says. "You multiply the numbers and keep the letter." Grandpa nods. Unlike adding, when you multiply you **can** join unlike terms together.

**In short:** To multiply terms, multiply the numbers and then write the letters, so $3x \times 2y = 6xy$ and $x \times x = x^2$. To divide, write it as a fraction and cancel what is the same on the top and bottom.

## Before you start

- No sign between a number and a letter means multiply: $4a = 4 \times a$.
- Powers: $a^3 = a \times a \times a$.
- Simplifying fractions by dividing top and bottom by the same number: $\frac{6}{9} = \frac{2}{3}$.
- Multiplying can be done in any order: $2 \times 7 \times 5 = 2 \times 5 \times 7 = 70$.

## Multiplying terms

### Key idea: Multiply numbers, then letters
1. Multiply all the **numbers** together.
2. Write all the **letters** after them in alphabetical order.
3. If a letter appears more than once, use a **power** to count how many times.

### Worked example: Simple products
$5 \times 4a = 5 \times 4 \times a = 20a$

$a \times b \times c = abc$

$5a \times 4b = 5 \times 4 \times a \times b = 20ab$

$a \times 10 \times b = 10ab$

Any terms can be multiplied, like or unlike. $3x \times 2y = 6xy$ is fine.

## Index notation

When the same letter is multiplied by itself, write it with a power.

$$x \times x \times x = x^3$$

$$x \times x \times 3y = 3x^2y$$

$$3 \times p \times p \times 5 = 15p^2$$

### Worked example: Powers times powers
$2a^2 \times 3a^3$

Numbers: $2 \times 3 = 6$

Letters: $a^2 \times a^3 = (a \times a) \times (a \times a \times a) = a^5$

Answer: $6a^5$

### Exam tip: Count the letters
When you multiply the same letter, you can **add the powers**: $a^2 \times a^3 = a^{2+3} = a^5$. Just remember that $a$ on its own is $a^1$. So $a \times a^4 = a^5$.

### Watch out: Adding is not multiplying
$x + x + x = 3x$ but $x \times x \times x = x^3$.

Adding counts **how many** $x$ there are. Multiplying uses a **power**.

## Dividing terms

A division can be written as a fraction. Then **cancel** anything that is on both the top and the bottom.

### Worked example: Cancel numbers and letters
$\frac{3a}{9a}$: the $a$ on top and bottom cancel. $\frac{3}{9} = \frac{1}{3}$. Answer: $\frac{1}{3}$

$\frac{10b^2}{5b} = \frac{10 \times b \times b}{5 \times b}$. Cancel one $b$, and $10 \div 5 = 2$. Answer: $2b$

### Worked example: Dividing a sum
$\frac{12b + 9c}{3}$: divide **each** term on top by 3.

$12b \div 3 = 4b$ and $9c \div 3 = 3c$

Answer: $4b + 3c$

### Watch out: Divide every term
In $\frac{12b + 9c}{3}$ you must divide both $12b$ and $9c$ by 3. Writing $4b + 9c$ is a common slip.

### Challenge: Selective level: mixed products and cancelling
Simplify $\frac{10a \times 5}{2a \times 20a}$.

Top: $10a \times 5 = 50a$.

Bottom: $2a \times 20a = 40a^2$.

$\frac{50a}{40a^2}$: cancel one $a$ and divide 50 and 40 by 10.

Answer: $\frac{5}{4a}$

Another trap: $p \times 7 - p \times 5$. Multiply first: $7p - 5p = 2p$.

**Mind map**

```text
Multiplying and dividing terms
  Multiply
    Numbers times numbers
    Letters in alphabetical order
    $5a \times 4b = 20ab$
  Index notation
    $x \times x \times x = x^3$
    Same letter: add the powers
  Divide
    Write as a fraction
    Cancel top and bottom
    $\frac{10b^2}{5b} = 2b$
  Divide a sum
    Divide every term
```

## Quick summary

- To multiply terms, multiply the numbers, then write the letters.
- Unlike terms **can** be multiplied: $3x \times 2y = 6xy$.
- Repeated letters use powers: $x \times x = x^2$.
- When multiplying the same letter, add the powers: $a^2 \times a^3 = a^5$.
- To divide, write a fraction and cancel what is on top and bottom.
- When the top is a sum, divide every term.

## Quiz

Try these questions from the chapter quiz.

1. $a \times 10 \times b = $
   - A. $a10b$
   - B. $10ab$
   - C. $10 + ab$
   - D. $11ab$
   - Answer: B. $10ab$. Multiply the numbers (just 10), then write the letters in order.

2. $6m \times 5n = $
   - A. $11mn$
   - B. $30m + n$
   - C. $30mn$
   - D. $65mn$
   - Answer: C. $30mn$. $6 \times 5 = 30$, then the letters $m$ and $n$.

3. $x \times x \times x \times x = $
   - A. $4x$
   - B. $x^4$
   - C. $x + 4$
   - D. $4x^4$
   - Answer: B. $x^4$. $x$ multiplied by itself 4 times is $x^4$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/multiplying-and-dividing-terms/#quiz