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Cubed Powers STRAND: Algebra TOPIC: Equations Cubed powers LITERACY TASK • Find the meanings of these key terms: 1. square number:................................................................................................................................................................. 2. cubic number: ................................................................................................................................................................... 3. equation: ............................................................................................................................................................................ 4. expression: ......................................................................................................................................................................... A square number is any number that can be arranged in a square. Multiplying each whole number (1, 2, 3… etc) by itself will generate this pattern of square numbers. 1. List the first 20 square numbers. In this list we can see that 9 + 16 = 25. We can write this as: 32 + 42 = 52. 2. Find two more pairs of square numbers, that when added together, produce another square number. Use these numbers to write an equation of the form: x2 + y2 = w2. A cubic number is any number that can be arranged in a cube. Cubic numbers are found by raising each whole number to the power of three. 3. List the first 20 cubic numbers. 4. Leonhard Euler proved there is no pair of cubic numbers that will add to another cubic number. From your list of twenty cubic numbers find two ‘near misses’ (i.e. the sum of the two cubes is one number off being a third cube). RESEARCH TASK ONE Research the life and works of Leonhard Euler. 5. From your cubic list, find three cubic numbers that add to a fourth cubic number. Use these numbers to write an equation of the form: x3 + y3 + z3 = w3. There are many sets of three cubic numbers that will add to a fourth cubic number. This situation can be represented by the equation: x3 + y3 + z3 = w3 with a solution that can be represented by the following algebraic equations: x = k3 + 1 y = 2k3 – 1 z = k(k3 – 2) w = k(k3 + 1) where k can be any positive integer. 6. Substitute your values of x, y, z and w found in part 5 into these equations. Now solve one of them to find the value of k. You should get the same answer to each equation so check the result by substituting your value of k into the other three equations. 7. Write the equation of the form: x3 + y3 + z3 = w3 generated when k = 3. Check this result with your calculator. 8. Find the equation generated when k = 5. 9. Find the value of k for the expression: 3343 + 6853 + 23873 = 24083. The equation x3 + y3 + z3 = w3 can also be solved using the following equations: x = 3k2 y = 6k2 – 3k + 1 z = 3k(3k2 – 2k + 1) – 1 w = z + 1 John Wiley & Sons Australia, Ltd 2001 STRAND: Algebra TOPIC: Equations INVESTIGATION: Cubed powers 10. Use k = 4 to find an equation of the form: x3 + y3 + z3 = w3. 11. Find the value of k that generates the expression: 753 + 1363 + 9893 = 9903. 12. Add any three consecutive cubes. What type of number is always formed? What is the sum of three consecutive cubes always divisible by? 13. Some cubes can be written as the sum of three other cubes in more than one way. Find an example of such a number. 14. 1083 can be written as the sum of three cubes in five different ways. Complete the following: (a) 1083 = 123 + ___ + ___. (b) 1083 = 133 + ___ + ___. (c) 1083 = 153 + ___ + ___. (d) 1083 = 243 + ___ + ___. (e) 1083 = 543 + ___ + ___. John Wiley & Sons Australia, Ltd 2001.
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