Question 10.9.2

We know from Question 7.4.9(b) that \[(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3,\] but before mathematicians developed algebra, they came up with geometric explanations for facts like this. For example, the Renaissance mathematician Cardano would have visualised $(a+b)^3$ as the volume of a cube with side length $a+b$. The cube can be cut up along the lines dividing each edge into $a$ and $b$:

1. How many pieces will the cube be cut into? You might like to try to draw the pieces. Show answer

   There are $8$ pieces (each edge is cut into $2$, so there should be $2\times2\times2$ pieces). You might describe them as a small cube, a large cube, three flat pieces, and three long pieces.

   

2. What is the volume of each of the pieces? Hence show that $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$. Show answer

   The small cube has volume $a^3$, the long pieces each have volume $a^2b$, the flat pieces each have volume $ab^2$, and the large cube has volume $b^3$. So the total volume is $a^3 + 3a^2b + 3ab^2 + b^3$. This must be equal to the volume of the original cube of side length $a+b$, which was $(a+b)^3$.