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NUMBER AND ALGEBRA • NUMBER AND PLACE VALUE ■ ■ ■ ■ The first cubic number, 1, equals 1 ì 1 ì 1. The second cubic number, 8, equals 2 ì 2 ì 2. The third square number, 27, equals 3 ì 3 ì 3. If this pattern is continued, any cubic number can be found by writing the position of the cubic number 3 times and multiplying. This is known as cubing a number and is written by an index (or power) of 3. For example, 43 = 4 ì 4 ì 4 = 64. WORKED EXAMPLE 14 Find the value of 53. THINK WRITE 1 Write 53 as the product of three lots of 5. 2 Evaluate. 53 = 5 ì 5 ì 5 = 125 Cube roots ■ ■ ■ ■ The opposite of cubing a number is finding the cube root of a number. The cube root is found by looking for a number that can be written three times and multiplied to produce the given number. The cube root symbol is similar to the square root symbol but with a small 3 written in front, as shown: 3 . From the worked example, we can see that 3 125 = 5. WORKED EXAMPLE 15 Find 3 27 . THINK WRITE 27 = 3 ì 3 ì 3 Look for a number that when written three times and multiplied gives 27. 3 27 = 3 REMEMBER 1. When a number is written three times and multiplied, this is called cubing a number and is written using an index (or power) of 3. 2. The opposite of cubing a number is finding the cube root of a number. This is calculated using our knowledge of cubic numbers or using a calculator. EXERCISE 3D Cubes and cube roots FLUENCY 1 WE14 Find the value of 43. 2 Find the value of: a 23 b 33 3 Write the first 10 cubic numbers. 4 WE15 Find 3 8 . 82 Maths Quest 7 for the Australian Curriculum c 63 d 103 NUMBER AND ALGEBRA • NUMBER AND PLACE VALUE INDIVIDUAL PATHWAYS eBoo k plus eBook Activity 3-D-1 Cubes and cube roots doc-1684 Activity 3-D-2 More cubes and cube roots doc-1685 Activity 3-D-3 Advanced cubes and cube roots doc-1686 UNDERSTANDING 5 Find each of the following. Verify your answers with a calculator. a 3 64 b 3 216 c 3 343 3 d 729 (Hint: Use your answer to question 3.) REASONING 6 The first 5 square numbers are 1, 4, 9, 16, 25. If we find the difference between these numbers, we get 4 – 1 = 3, 9 – 4 = 5, 16 – 9 = 7 and 25 – 16 = 9. These numbers all differ by 2. Representing this in a table, we get: Square numbers First difference Second difference eBoo k plus eBook Digital doc WorkSHEET 3.2 doc-1688 1 4 3 9 5 2 16 7 2 25 9 2 Repeat this process for the first 6 cubic numbers. How many times did you need to find the difference until they were equal? If you look at 14, 24, 34, 44, . . ., how many differences would you need to find until it they were equal? REFLECTION What would be the first 4 numbers that could be arranged as a triangle-based pyramid (all triangles equilateral)? Chapter 3 Indices and primes 83