Cubes and Cube Roots

The cube of a number is the number multiplied by itself three times, so 2^3 = 8 and 5^3 = 125. Numbers such as 1, 8, 27, 64 and 125 are called perfect cubes because each is the volume of a solid cube of unit cubes. In this chapter you will study the famous Hardy-Ramanujan number 1729, which can be written as the sum of two cubes in two ways, and discover patterns in cubes: the cube of an odd number is odd, the units digit of a cube depends only on the units digit of the number, and the sums 1, 3 + 5, 7 + 9 + 11 give successive cubes. You will use prime factorisation to test whether a number is a perfect cube, since its prime factors must form groups of three, and to find the smallest number by which a given number must be multiplied or divided to make it a perfect cube. Finally, you will find cube roots by prime factorisation and estimate the cube root of a large perfect cube by grouping digits.

Detailed Worksheet: Cubes and Cube Roots Section A - Definitions (10 marks) 1. What is a perfect cube? Write the first six perfect cubes of natural numbers. (2 marks) 2. Define the cube root of a number. Find the cube root of 343 and of -8. (2 marks) 3. Why is 1729 called the Hardy-Ramanujan number? Show the two ways in which it can be written as a sum of two cubes. (2 marks) 4. State the condition on the prime factorisation of a number for it to be a perfect cube. Illustrate with 216. (2 marks) 5. Write the units digit of the cube of a number whose units digit is (i) 2 (ii) 3 (iii) 7 (iv) 8. What do you notice? (2 marks) Section B - Calculations and Applications (15 marks) 6. Is 243 a perfect cube? If not, find the smallest natural number by which it must be multiplied to make a perfect cube, and find the cube root of the product. (3 marks) 7. Find the cube roots of the following by prime factorisation: (i) 3375 (ii) 10648 (iii) 46656. (3 marks) 8. Find the smallest number by which 8640 must be divided so that the quotient is a perfect cube. Also find the cube root of the quotient. (3 marks) 9. Estimate the cube roots of the perfect cubes 857375 and 13824 by grouping digits, explaining each step. (3 marks) 10. Find the cubes of (i) 2/3 (ii) -0.3 (iii) 1.2. (3 marks) Section C - Diagrams (10 marks) 11. Draw a cube of edge 3 units built from unit cubes, showing the layers clearly. How many unit cubes does it contain? How many more unit cubes are needed to make a cube of edge 4 units? (4 marks) 12. Draw a factor tree for 2744. Group the prime factors in triples and find the cube root of 2744. (3 marks) 13. Write the pattern 1 = 1^3, 3 + 5 = 2^3, 7 + 9 + 11 = 3^3, 13 + 15 + 17 + 19 = 4^3 in the form of a triangle of odd numbers, and use it to write the next line for 5^3. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Parikshit makes a cuboid of plasticine of sides 5 cm, 2 cm and 5 cm. How many such cuboids will he need to form a cube? What will be the edge of that cube? Explain using prime factorisation. (5 marks) 15. State whether each statement is true or false, giving a reason or an example: (i) The cube of any odd number is even. (ii) A perfect cube does not end with two zeros. (iii) If the square of a number ends with 5, then its cube ends with 25. (iv) There is no perfect cube which ends with 8. (v) The cube of a two-digit number may be a three-digit number. (5 marks) 16. A cubical box has a volume of 21952 cm^3. Find the length of its edge by prime factorisation, and then its total surface area. If each edge is doubled, how many times does the volume become, and what is the new volume? (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show complete prime factorisation wherever it is used and draw diagrams neatly in pencil.