Step by step
The question
Question: expand (a + b)³.
Careful!
A power of a bracket is not the power of each part.
(a + b) to the power 3 is the bracket times itself 3 times.
There are middle terms too.
1 Use Pascal's triangle
Pascal's triangle starts with 1.
Every row starts and ends with 1.
Each other number is the two numbers above it added.
2 The pattern
There are 4 terms.
The numbers in front come from row 3: 1, 3, 3, 1.
The powers of a go down from 3 to 0.
The powers of b go up from 0 to 3.
Remember: any number to the power 0 is 1.
3 Work out each term
The first term: 1, a to the power 3, b to the power 0.
That is a³.
Second term: So 3a²b.
Third term: So 3ab².
Fourth term: So b³.
4 Add them up
So (a + b)³ is a³ + 3a²b + 3ab² + b³.
5 Check
Let's check with a = 1, b = 1.
The question: 2 to the power 3 is 8.
Our answer with a = 1: the terms add up to 8 too.
It works!
Let's look back.
First: use Pascal's triangle.
Then: the pattern.
Then: work out each term.
Then: add them up.
Last: check.