What is the side of the cube whose volume is p³ + 6p²q + 12pq² + 8q³ cubic units?

Step-by-step solution

Idea: Volume of a cube = (side)³. So write the volume in the form a³ + 3a²b + 3ab² + b³ = (a + b)³; then a + b is the side.

1. First and last terms: p³ = (p)³ and 8q³ = (2q)³. So try a = p, b = 2q.1 mark
2. Check the middle terms: 3a²b = 3(p)²(2q) = 6p²q ✓; 3ab² = 3(p)(2q)² = 12pq² ✓.1 mark
3. So the volume = (p)³ + 3(p)²(2q) + 3(p)(2q)² + (2q)³ = (p + 2q)³.½ mark
4. Side of the cube = p + 2q units.½ mark

Check: p = 1, q = 1: volume = 1 + 6 + 12 + 8 = 27 = 3³, and side p + 2q = 3 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Taking b = 8q or 2q³. The cube root of 8q³ is 2q.
• Not checking the two middle terms. Only if both match is the expression a perfect cube.
• Giving the answer in cubic units. A side is a length, so the unit is just ‘units’.

How this can come in the exam

Try one yourself

Find the side of the cube whose volume is 27m³ + 27m²n + 9mn² + n³.

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