Step by step
The question
Question: expand (x + y)⁵.
Careful!
A power of a bracket is not the power of each part.
(x + y) to the power 5 is the bracket times itself 5 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 6 terms.
The numbers in front come from row 5: 1, 5, 10, 10, 5, 1.
The powers of x go down from 5 to 0.
The powers of y go up from 0 to 5.
Remember: any number to the power 0 is 1.
3 Work out each term
The first term: 1, x to the power 5, y to the power 0.
That is x⁵.
Second term: So 5x⁴y.
Third term: So 10x³y².
Fourth term: So 10x²y³.
Fifth term: So 5xy⁴.
Sixth term: So y⁵.
4 Add them up
So (x + y)⁵ is x⁵ + 5x⁴y + 10x³y² + 10x²y³ + 5xy⁴ + y⁵.
5 Check
Let's check with x = 1, y = 1.
The question: 2 to the power 5 is 32.
Our answer with x = 1: the terms add up to 32 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.