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Numbers LESSON 2: PERFECT SQUARES AND CUBES, SQUARE AND CUBE ROOTS Todays Objectives Students will be able to demonstrate an understanding of factors of whole numbers by determining the: prime factors, greatest common factor (GCF), least common multiple (LCM), square root, cube root, including: Determine, concretely, whether a given whole number is a perfect square, a perfect cube or neither Determine, using a variety of strategies, the square root of a perfect square, and explain the process Determine, using a variety of strategies, the cube root of a perfect cube, and explain the process Perfect Squares and Perfect Cubes For whole numbers, the word perfect in the term perfect square means that the square can be written as a product of two identical whole numbers. Similarly, a perfect cube can be written as a product of three identical whole numbers. For example, 81 is a perfect square because it can be written as 9 x 9, and 8 is a perfect cube because it can be written as 2 x 2 x 2. A perfect square can be represented as the area of a square with whole number dimensions. Similarly, a perfect cube can be represented as the volume of a cube with whole number dimensions. Perfect Squares and Perfect Cubes 2 cm 2 cm Area = 4 square cm (cm2) 2 cm Volume = 8 cubic cm (cm3) 2 cm 2 cm Perfect Square Perfect Cube Perfect Squares and Perfect Cubes If you have square tiles, you could use them to determine or show that a number is a perfect square by actually forming a square with the tiles. If you have some small cubes, such as dice, you can use them to determine or show that a number is a perfect cube by constructing a larger cube out of the smaller ones. Example How could you use 36 square tiles to show that the number 36 is a perfect square? Solution: You could construct a square using all 36 tiles. The resulting square will have dimensions 6 tiles x 6 tiles. Thus, 6 x 6 = 36, and 36 is a perfect square How could you use 30 sugar cubes to show that the number 30 is NOT a perfect cube? Solution: You could attempt to construct a cube out of the sugar cubes and prove that it is impossible, therefor proving that 30 is NOT a perfect cube. Square Roots and Cube Roots The square root of a number, x, is the number, s, such that s x s = x. For example, the square root of 4, is 2 (2 x 2 = 4) The cube root of a number, x, is the number q, such that q x q x q = x. For example, the cube root of 8 is 2 (2 x 2 x 2 = 8) You should be able to recognize perfect squares and cubes that are 100 or less, and, consequently, know their square and cube roots. For example, you should be able to recognize 49 as a perfect square (49 = 7 x 7) and 27 as a perfect cube (3 x 3 x 3 = 27). Knowing these allows you to state that the positive square root of 49 is 7, and the cube root of 27 is 3. Positive and Negative Roots The symbol that we use to show the operation of taking the positive square root is √, and the symbol for cube root is 3√. These symbols are called radical signs. The symbol √ represents the positive square root so that √49 = 7. The negative square root is represented by -√ so that, for example, -√49 = -(7) = -7. The symbol for both the positive and negative square roots is ±√ so that, for example, ±√49 = ±7. The square root of a negative number is not a real number so that, for example, √-49 does not exist as a real number. There are however cube roots of positive and negative numbers so that , for example, 3√27 = 3 and 3√-27 = -3. Square Roots of Perfect Squares and Cube Roots of Perfect Cubes You should be able to recognize perfect squares and perfect cubes that are relatively small numbers, such as the ones shown in the following chart. Perfect Squares Perfect Cubes 1 = 12 25 = 52 81 = 92 8 = 23 4 = 22 36 = 62 100 = 102 27 = 33 9 = 32 49 = 72 121 = 112 125 = 53 16 = 42 64 = 82 144 = 122 1000 = 103 Being able to recognize basic perfect squares and perfect cubes, can help you find or estimate square and cube roots without using a calculator. Example Determine the positive square root of 3969 without using a calculator Solution: First, recognize that 6 x 6 = 36, so that 60 x 60 = 3600, which is smaller than 3969. Also, 70 x 70 = 4900, which is larger than 3969. Thus, the square root of 3969 is between 60 and 70, and likely closer to 60, because 3600 is closer to 3969 when compared to 4900. Notice that the ones digit in 3969 is 9, and therefore, if 3969 is a perfect square, the ones digit in its square root must be a 3 or 7. Why? 3 and 7 are the only single-digit numbers whose squares have a ones digit of 9. Thus, try 63 as the possible square root by multiplying it by itself: 63 x 63 = 3969, therefore, √3969 is 63. Example (You do) Determine the positive square root of 1156 without using a calculator, and explain the process you used. Solution: 3 x 3 = 9, so 30 x 30 = 900. Also, 4 x 4 = 16, so 40 x 40 = 1600. Answer is likely closer to 30, because 900 is closer to 1156 than 1600. The ones digit is 6, so ones digit of square root must be 4 or 6….try 34 and 36. 34 x 34 = 1156, therefore, √1156 = 34. Example (You do) Determine the cube root of 12 167 without using a calculator. Solution: 103 = 1000, 203 = 8000, 303 = 27000, so the cube root of 12167 is between 20 and 30, and closer to 20. The ones digit is 7, so if 12167 is a perfect cube, the ones digit must be a 3. So try 23 as the cube root. 23 x 23 = 529 529 x 23 = 12167 Therefore, 3√12167 = 23. Wall Quiz! Teams of 3 When I say go, your team should move around to the different questions located on the walls You cannot bring a calculator! Try to answer each question, and record your answers on a piece of paper The team that gets the most questions right in the given time limit will win candy!