This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems. 1. Write the unit digit in the cube of the following numbers: To find the unit digit of a cube, we only need to cube the unit digit of the original number and find its unit digit. a) 19 Step 1: The unit digit of 19 is 9. Step 2: Cube the unit digit: 9^3 = 729. Step 3: The unit digit of 729 is 9. The unit digit is 9. b) 201 Step 1: The unit digit of 201 is 1. Step 2: Cube the unit digit: 1^3 = 1. Step 3: The unit digit of 1 is 1. The unit digit is 1. c) 164 Step 1: The unit digit of 164 is 4. Step 2: Cube the unit digit: 4^3 = 64. Step 3: The unit digit of 64 is 4. The unit digit is 4. d) 1307 Step 1: The unit digit of 1307 is 7. Step 2: Cube the unit digit: 7^3 = 343. Step 3: The unit digit of 343 is 3. The unit digit is 3. e) 510 Step 1: The unit digit of 510 is 0. Step 2: Cube the unit digit: 0^3 = 0. Step 3: The unit digit of 0 is 0. The unit digit is 0. f) 35 Step 1: The unit digit of 35 is 5. Step 2: Cube the unit digit: 5^3 = 125. Step 3: The unit digit of 125 is 5. The unit digit is 5. g) 716 Step 1: The unit digit of 716 is 6. Step 2: Cube the unit digit: 6^3 = 216. Step 3: The unit digit of 216 is 6. The unit digit is 6. h) 1718 Step 1: The unit digit of 1718 is 8. Step 2: Cube the unit digit: 8^3 = 512. Step 3: The unit digit of 512 is 2. The unit digit is 2. 2. State which of the following numbers are perfect cubes: A number is a perfect cube if it can be expressed as n^3 for some integer n. a) 1000 Step 1: Find the cube root of 1000. [3]1000 = 10 Step 2: Since 1000 is the cube of an integer (10), it is a perfect cube. Perfect cube b) 5^3 × 7^3 Step 1: Use the property (a × b)^n = a^n × b^n. 5^3 × 7^3 = (5 × 7)^3 = 35^3 Step 2: Since the number can be expressed as the cube of an integer (35), it is a perfect cube. Perfect cube c) 100000 Step 1: Count the number of zeros. There are 5 zeros. Step 2: For a number ending in zeros to be a perfect cube, the number of zeros must be a multiple of 3. Since 5 is not a multiple of 3, 100000 is not a perfect cube. Not a perfect cube d) (100)^2 × 10 Step 1: Simplify the expression. (100)^2 × 10 = 10000 × 10 = 100000 Step 2: As determined in part (c), 100000 is not a perfect cube. Not a perfect cube e) (1000)^2 Step 1: Simplify the expression. (1000)^2 = 1000 × 1000 = 1000000 Step 2: Find the cube root of 1000000. [3]1000000 = 100 Step 3: Since 1000000 is the cube of an integer (100), it is a perfect cube. Perfect cube 3. Find the cube of the following numbers: a) 21 Step 1: Calculate 21^3. 21^3 = 21 × 21 × 21 = 441 × 21 = 9261 The cube of 21 is 9261. b) 100 Step 1: Calculate 100^3. 100^3 = 100 × 100 × 100 = 1000000 The cube of 100 is 1000000. c) 17 Step 1: Calculate 17^3. 17^3 = 17 × 17 × 17 = 289 × 17 = 4913 The cube of 17 is 4913. d) 11^3 Step 1: Calculate 11^3. 11^3 = 11 × 11 × 11 = 121 × 11 = 1331 The cube of 11 is 1331. 4. Find the sum of consecutive odd numbers to obtain the following cubes: The sum of n consecutive odd numbers starting from n^2 - n + 1 equals n^3. a) 4^3 Step 1: Identify n. Here n=4. Step 2: Find the first odd number in the sequence: n^2 - n + 1 = 4^2 - 4 + 1 = 16 - 4 + 1 = 13. Step 3: Write the sum of 4 consecutive odd numbers starting from 13. 4^3 = 13 + 15 + 17 + 19 = 64 The sum is 13 + 15 + 17 + 19. b) 7^3 Step 1: Identify n. Here n=7. Step 2: Find the first odd number in the sequence: n^2 - n + 1 = 7^2 - 7 + 1 = 49 - 7 + 1 = 43. Step 3: Write the sum of 7 consecutive odd numbers starting from 43. 7^3 = 43 + 45 + 47 + 49 + 51 + 53 + 55 = 343 The sum is 43 + 45 + 47 + 49 + 51 + 53 + 55. c) 8^3 Step 1: Identify n. Here n=8. Step 2: Find the first odd number in the sequence: n^2 - n + 1 = 8^2 - 8 + 1 = 64 - 8 + 1 = 57. Step 3: Write the sum of 8 consecutive odd numbers starting from 57. 8^3 = 57 + 59 + 61 + 63 + 65 + 67 + 69 + 71 = 512 The sum is 57 + 59 + 61 + 63 + 65 + 67 + 69 + 71. d) 11^3 Step 1: Identify n. Here n=11. Step 2: Find the first odd number in the sequence: n^2 - n + 1 = 11^2 - 11 + 1 = 121 - 11 + 1 = 111. Step 3: Write the sum of 11 consecutive odd numbers starting from 111. 11^3 = 111 + 113 + 115 + 117 + 119 + 121 + 123 + 125 + 127 + 129 + 131 = 1331 The sum is 111 + 113 + 115 + 117 + 119 + 121 + 123 + 125 + 127 + 129 + 131. Send me the next one 📸