Cubing a number
To cube a number, multiply three copies of it together. 3 cubed, written , is 3 × 3 × 3 = 27. Build a cube 3 small cubes long, 3 wide and 3 high and it holds 27 small cubes: 3 × 3 = 9 in each layer, and 3 layers.
The cube numbers begin 1, 8, 27, 64 and 125, from up to , and .
A cube with edge 3 holds 3 layers of 9 small cubes: .
The cube root
The cube root goes back from the cube number to the number that was cubed. The cube root of 27, written , is the number that makes 27 when three copies of it are multiplied together. 3 × 3 × 3 = 27, so is 3.
Cubing and taking the cube root undo each other: , and is 5.
Three copies of 3 multiply to 27, so the cube root of 27 is 3.
Two copies or three
The square root and the cube root ask different questions. The square root of 64 asks which number makes 64 from two copies: 8 × 8 = 64, so is 8. The cube root of 64 asks which number makes 64 from three copies: 4 × 4 × 4 = 64, so is 4.
64 tiles make a flat square with side 8, so the square root of 64 is 8.
The same 64 make a cube with edge 4, so the cube root of 64 is 4.
The edge of a cube
The volume of a cube is edge × edge × edge, the cube of its edge. So when the volume is known, the cube root gives the edge. A cube that holds 64 unit cubes has an edge of , which is 4, and every edge is the same length.
When a number is not a cube number, its cube root lies between two whole numbers. 50 lies between the cube numbers 27 and 64, so is between 3 and 4.
The usual mistakes
Cubing is not multiplying by 3. is 4 × 4 × 4 = 64, not 4 × 3 = 12.
Cubing needs three copies, not two. 4 × 4 = 16 is , not .
The cube root is not a third. is 3, not 27 ÷ 3 = 9. Check by cubing: 9 × 9 × 9 = 729, far more than 27.
Worked example: A Square Plot of Known Area and a Cube-Shaped Tank of Known Volume
Question A farmer has a square vegetable plot with an area of 196 m2 and a cube-shaped water tank with a volume of 343 m3. (a) Find the length of one side of the plot and the length of fencing needed to go once round it. (b) Find the length of one edge of the tank.
1.Let the side of the plot be s m. The area of a square is the side times the side, so s2 = 196 and s = √196. The number −14 also has a square of 196, but a length cannot be negative, so only the positive root is used.
The area of a square is its side squared, so s2 = 196 and s = √196. 2.102 = 100 and 202 = 400, so s lies between 10 and 20. A square that ends in 6 comes from a number that ends in 4 or in 6. Try 14: 14 × 14 = 196. So s = 14.
s lies between 10 and 20, and a square that ends in 6 comes from 14 or 16: 14 × 14 = 196. 3.(a) One side of the plot is 14 m. The fencing goes along all four sides, so its length is 4 × 14 = 56 m.
(a) One side is 14 m, and the fencing is 4 × 14 = 56 m long. 4.Let the edge of the tank be e m. The volume of a cube is edge × edge × edge, so e3 = 343 and e is the cube root of 343. 53 = 125 and 103 = 1000, so e lies between 5 and 10. Only a number that ends in 7 has a cube that ends in 3, so try 7.
The volume of a cube is its edge cubed, so e3 = 343. The edge lies between 5 and 10. 5.(b) 7 × 7 × 7 = 49 × 7 = 343, so one edge of the tank is 7 m long.
(b) 7 × 7 × 7 = 343, so one edge of the tank is 7 m long.
Answer: (a) side 14 m, fencing 56 m; (b) 7 m
Common mistakes
- Dividing the area by 4, 196 ÷ 4 = 49, to find the side. Dividing by 4 finds a side from the perimeter. The area is the side multiplied by itself, so the side is the square root, 14 m.
- Dividing the volume by 3 to find the edge. The edge is multiplied by itself three times, not added three times, so the edge is the cube root of 343, which is 7 m.