In algebra we use letters to represent unknown numbers.

When we are writing algebra we do not use a times sign.

We can write a × b as ab

We can write 5 × c as 5c

We can write 6 × a × b as 6ab

When we have the same letter multiplied together we can use a power.

In the same way that we say the 5 × 5 = 5² we can say that a × a = a²

b × b × b = b^{3}

p × p × p × p × p = p^{5}

**Example 1: Simplify: 4a × 5b**

We can multiply the 4 and 5 together

4 × 5 = 20

We now have 20 × a × b

We don't write the times sign in algebra.

We write 20 × a × b as 20ab

**The answer is 20ab**

**Example 2: Simplify: 10t × 3t**

We can multiply the 10 and 3 together

10 × 3 = 30

We now have 30 × t × t

t × t = t^{2}

30 × t^{2} = 30t^{2}

**The answer is 30t ^{2}**

We write division as a fraction

We write a ÷ b as

**Example 3: Simplify 12a**

We can look at this question in two parts:

We have 12 ÷ 4

12 ÷ 4 = 3

We also have a^{3} ÷ a

a^{3} ÷ a = a^{2}

**The answer is 3a ^{2}**

**Try these:**

We can also simplify algebra by collecting like terms (adding up, or subtracting, things that are the same).

In the same way that 5 + 5 + 5 can be written as 3 × 5

a + a + a can be written as 3 × a or 3a

In the same way that 5 + 5 + 5 + 5 can be written as 4 × 5

a + a + a + a can be written as 4 × a or 4a

We can add 3a and 4a

a + a + a + a + a + a + a = 7a

3a + 4a = 7a

We can add b terms to other b terms, and subtract b terms from other b terms:

10b + 6b = 16b

2b + 5b = 7b

8b + 2b = 10b

7b - 4b = 3b

10b - 8b = 2b

We cannot add 3a and 4b because they are not like terms (we can only add a terms to other a terms).

**Example 4: Simplify: 3a + 5b + 7a + 4b**

We can add the a term to the other a term and the b term to the other b term.

3a + 5b + 7a + 4b

For the a terms we have: 3a + 7a

3a + 7a = 10a

For the b terms we have 5b + 4b

5b + 4b = 9b

The answer is 10a + 9b

**Example 5: Simplify: 8a + b - 2a + 4b**

We collect the a terms and collect the b terms.

Each term always goes with the sign in front of it.

8a + b - 2a + 4b

For the a terms we have: 8a - 2a

8a - 2a = 6a

For the b terms we have b + 4b

b + 4b = 5b

The answer is 6a + 5b

**Example 6: Simplify: 5a - 3ab - 9a - ab**

This time we have a terms and ab terms.

We can collect the a terms and collect the ab terms.

Each term takes the sign from in front of it.

5a - 3ab - 9a - ab

For the a terms we have: 5a - 9a

5a - 9a = -4a

For the ab terms we have -3ab - ab

-3ab - ab = -4ab

The answer is -4a - 4ab

**Try these:**