What if we have a cube of edge a + b? Can we divide a cube of edge (a + b) into smaller cubes and cuboids and represent this new identity?

Step-by-step solution

Idea: “This new identity” is (a + b)³ = a³ + 3a²b + 3ab² + b³. Just as a square of side a + b splits into a², two ab rectangles and b², a cube of edge a + b splits into pieces whose volumes add up to the same total.

1. Cut the cube with three planes: one across the width, one across the height and one across the depth, each at distance a from a corner. Every edge is now split into a and b, and the cube falls into 2 × 2 × 2 = 8 pieces.½ mark
2. Each piece has three dimensions, each either a or b. Sort them by how many a‘s they have.½ mark
3. Three a‘s: one cube a × a × a, volume a³. Three b‘s: one cube b × b × b, volume b³.½ mark
4. Two a‘s and one b: three cuboids a × a × b (the b can be the width, the height or the depth), total 3a²b.½ mark
5. One a and two b‘s: three cuboids a × b × b, total 3ab².½ mark
6. The 8 pieces fill the whole cube, so (a + b)³ = a³ + 3a²b + 3ab² + b³.½ mark

Check: Count pieces: 1 + 3 + 3 + 1 = 8 = 2 × 2 × 2 ✓. Numbers: a = 3, b = 2: 125 = 27 + 54 + 36 + 8 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Counting only one aab-cuboid. The short side b can point in any of 3 directions, so there are 3 of them (and 3 of the abb kind).
• Thinking there are 6 pieces. Splitting all three edges gives 2 × 2 × 2 = 8 pieces.
• Calling the cuboids cubes. Only the pieces with all edges equal (a, a, a or b, b, b) are cubes.

How this can come in the exam

Try one yourself

A cube of edge 10 cm is cut into pieces by splitting each edge as 8 cm + 2 cm. Find the total volume of the three largest cuboids (not cubes).

More questions like this

• What happens when we replace b with −b in this new identity?
• What is the side of the cube whose volume is p³ + 6p²q + 12pq² + 8q³ cubic units?
• Now consider the expression 8n³ − 60n²m + 150nm² − 125m³. If you write it in the form (a − b)³, what will be a and b?
• Now let us play with known identities to discover more identities. Try to multiply the following using the distributive property.
• We already know that x² − y² = (x − y)(x + y).
  Further, we have verified that x³ − y³ = (x − y)(x² + xy + y²).
  Observe that x − y is a common factor of x² − y² and x³ − y³.
  Do you think x − y is also a factor of x⁴ − y⁴?
  Note that x⁴ − y⁴ = (x²)² − (y²)² = (x² − y²) (x² + y²).
  Can you see how x − y is a factor of x⁴ − y⁴?
  How about x⁵ − y⁵? Does this also have x − y as a factor?

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