Step by step
The question
Question: expand (a + b)⁴.
Careful!
A power of a bracket is not the power of each part.
(a + b) to the power 4 is the bracket times itself 4 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 5 terms.
The numbers in front come from row 4: 1, 4, 6, 4, 1.
The powers of a go down from 4 to 0.
The powers of b go up from 0 to 4.
Remember: any number to the power 0 is 1.
3 Work out each term
The first term: 1, a to the power 4, b to the power 0.
That is a⁴.
Second term: So 4a³b.
Third term: So 6a²b².
Fourth term: So 4ab³.
Fifth term: So b⁴.
4 Add them up
So (a + b)⁴ is a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴.
5 Check
Let's check with a = 1, b = 1.
The question: 2 to the power 4 is 16.
Our answer with a = 1: the terms add up to 16 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.