Question

Prove the identity (a + b)^2 = a ^2 + 2ab + b^2 for all natural numbers a, b.

Answer 1

Explanation:

To prove the identity we just manually compute the left hand side of it, simplify it and check that we do get the right hand side of it:

`(a+b)^2=(a+b)\cdot(a+b)` (that's the definition of squaring a number)

`=a\cdot a + a\cdot b + b \cdot a +b \cdot b` (we distribute the product)

`=a^2+ab+ba+b^2` (we just use square notation instead for the first and last term)

`=a^2+ab+ab+b^2` (since product is commutative, so that ab=ba)

`=a^2+2ab+b^2` (we just grouped the two terms ab into a single term)