A cube in eight pieces
Identity for (a + b)³
Cut the cube
A cube has edge a + b. Cut it with one plane across each direction, at the joins: one plane from the top, one from the front and one from the side.
Into how many pieces does the cube fall?
What this lesson covers
The idea
(a + b)³ = a³ + 3a²b + 3ab² + b³; a cube of edge a + b splits into cubes of volume a³ and b³ and six cuboids, three of volume a²b and three of volume ab².
Cut the cube
A cube has edge a + b. Cut it with one plane across each direction, at the joins: one plane from the top, one from the front and one from the side.
Into how many pieces does the cube fall?
- 4
- 8
- 12
Pull it apart
Pull the pieces apart with the slider. Tap each kind of piece to light it up and count it: the big cube, the small cube and the two kinds of cuboid.
Two cubes, six cuboids
(a + b)³ = a³ + 3a²b + 3ab² + b³. A cube of edge a + b splits into a cube a³, a cube b³, three cuboids a²b and three cuboids ab².
The algebra agrees: (a + b)³ = (a + b)(a² + 2ab + b²) = a³ + 2a²b + ab² + a²b + 2ab² + b³ = a³ + 3a²b + 3ab² + b³.
Notes
(a + b)³ = a³ + 3a²b + 3ab² + b³: a cube of edge a + b is two cubes and six cuboids.
Check yourself
Take a = 2 and b = 1. The three cuboids of size a × a × b hold 3a²b. What is 3a²b?
Answer: 12
One cuboid holds 4, and there are three of them: 12.
A cube has edge x + 1. Which is its volume (x + 1)³?
Find 11³ using (10 + 1)³.
Answer: 1331
1000 + 300 + 30 + 1 = 1331.
A cube has volume p³ + 6p²q + 12pq² + 8q³. What is the length of its edge?
- x³ + 3x² + 3x + 1 — correct. Yes: a = x and b = 1 give a³ = x³, 3a²b = 3x², 3ab² = 3x and b³ = 1.
- x³ + 1. The six cuboids are missing.
- x³ + 6x + 1. There are six cuboids, but of two kinds: three of volume x² and three of volume x.
- p + 8q. 8q³ is b³, so b = 2q. The cube root of 8 is 2.
- p + 2q — correct. Yes: p³ + 3(p)²(2q) + 3(p)(2q)² + (2q)³ = (p + 2q)³.
- p + 6q. 6 is a coefficient. Match 3a²b = 6p²q with a = p to find b = 2q.