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 = b3
p × p × p × p × p = p5
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 = t2
30 × t2 = 30t2
The answer is 30t2
We write division as a fraction
We write a ÷ b as
Example 3: Simplify
We can look at this question in two parts:
We have 12 ÷ 4
12 ÷ 4 = 3
We also have a3 ÷ a
a3 ÷ a = a2
The answer is 3a2
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: