Step by step
The question
Question: expand (m − n)⁴.
Careful!
A power of a bracket is not the power of each part.
(m − n) 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 m go down from 4 to 0.
The powers of −n go up from 0 to 4.
Remember: any number to the power 0 is 1.
3 Work out each term
The first term: 1, m to the power 4, −n to the power 0.
That is m⁴.
Second term: 4 times −1 is −4.
So −4m³n.
Third term: So 6m²n².
Fourth term: 4 times −1 is −4.
So −4mn³.
Fifth term: So n⁴.
4 Add them up
So (m − n)⁴ is m⁴ − 4m³n + 6m²n² − 4mn³ + n⁴.
5 Check
Let's check with m = 1, n = 1.
The question: 0 to the power 4 is 0.
Our answer with m = 1: the terms add up to 0 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.