mathematics Q2

• What is a perfect square?


  A perfect square is a number that can be expressed as the product of two equal integers.
• What is a perfect cube?


  A perfect cube is a number that can be expressed as the product of three equal integers.
• What are the properties of perfect squares?


  • A perfect square is a product of two equal integers.
  • The square root of a perfect square is an integer.
  • The last digit can only be 0, 1, 4, 5, 6, or 9.
  • The sum of the digits is always divisible by 3.
• What is an example of a perfect square?


  25 is a perfect square because it can be expressed as 5 x 5.
• Why is 49 a perfect square?


  49 is a perfect square because it can be expressed as 7 x 7, and its square root is 7, which is an integer.
• What are the properties of perfect cubes?


  • A perfect cube is a product of three equal integers.
  • The cube root of a perfect cube is an integer.
  • The last digit can only be 0, 1, 8, or 9.
  • The sum of the digits is always divisible by 9.
• What is an example of a perfect cube?


  27 is a perfect cube because it can be expressed as 3 x 3 x 3.
• Why is 64 a perfect cube?


  64 is a perfect cube because it can be expressed as 4 x 4 x 4, and its cube root is 4, which is an integer.
• What are some examples of perfect squares?


  • 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, etc.
• Which number is both a perfect square and a perfect cube?


  1 is both a perfect square and a perfect cube.
• How can you identify if a number is a perfect square?


  Check if the square root is an integer, the last digit is 0, 1, 4, 5, 6, or 9, and the sum of the digits is divisible by 3.
• What are some examples of perfect cubes?


  • 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, etc.
• How can you identify if a number is a perfect cube?


  Check if the cube root is an integer, the last digit is 0, 1, 8, or 9, and the sum of the digits is divisible by 9.
• Is the number 81 a perfect square, a perfect cube, both, or neither?


  81 is both a perfect square and a perfect cube.
• What is the last digit of a perfect square?


  The last digit of a perfect square can only be 0, 1, 4, 5, 6, or 9.
• What is the last digit of a perfect cube?


  The last digit of a perfect cube can only be 0, 1, 8, or 9.
• What is the sum of the digits of a perfect square?


  The sum of the digits of a perfect square is always divisible by 3.
• What is the square root of 81?


  9
• What is the sum of the digits of a perfect cube?


  The sum of the digits of a perfect cube is always divisible by 9.
• Why is the last digit of 81 significant in determining if it is a perfect square?


  Because it is one of the valid last digits for a perfect square.
• What is the sum of the digits of 81, and why is it relevant?


  The sum is 9, which is divisible by 3, indicating it could be a perfect square.
• Why is the last digit of 81 significant in determining if it is a perfect cube?


  Because it is one of the valid last digits for a perfect cube.
• What is the sum of the digits of 81, and why is it relevant for perfect cubes?


  The sum is 9, which is divisible by 9, indicating it could be a perfect cube.
• What is the cube root of 81?


  3
• What are the properties that determine if a number is a perfect square or perfect cube?

  • Perfect square:
  • Square root is an integer.
  • Last digit is one of the valid last digits (0, 1, 4, 5, 6, 9).
  • Sum of digits is divisible by 3.
  • Perfect cube:
  • Cube root is an integer.
  • Last digit is one of the valid last digits (0, 1, 8).
  • Sum of digits is divisible by 9.
• Is the number 121 a perfect square, a perfect cube, both, or neither?


  Perfect square
• What is the square root of 121?


  11
• Why is the last digit of 121 significant in determining if it is a perfect square?


  Because it is one of the valid last digits for a perfect square.
• What is the sum of the digits of 121, and why is it relevant?


  The sum is 4, which is not divisible by 3, indicating it is not a perfect square.
• Is the number 1000 a perfect square, a perfect cube, both, or neither?


  Perfect cube
• What is the square root of 1000?


  Approximately 31.62
• Why is the last digit of 1000 significant in determining if it is a perfect square?


  Because it is one of the valid last digits for a perfect square.
• What is the sum of the digits of 1000, and why is it relevant?


  The sum is 1, which is divisible by 3, but the square root is not an integer, so it is not a perfect square.
• Is the number 216 a perfect square, a perfect cube, both, or neither?


  Perfect cube
• What is the cube root of 216?


  6
• What is the square root of 216?


  Approximately 14.83
• Why is the last digit of 216 significant in determining if it is a perfect cube?


  Because it is one of the valid last digits for a perfect cube.
• What is the sum of the digits of 216, and why is it relevant?


  The sum is 9, which is divisible by 9, indicating it is a perfect cube.
• What are the criteria for a number to be classified as a perfect square or perfect cube?

  • Perfect square:
  • Square root must be an integer.
  • Last digit must be one of the valid last digits (0, 1, 4, 5, 6, 9).
  • Sum of digits must be divisible by 3.
  • Perfect cube:
  • Cube root must be an integer.
  • Last digit must be one of the valid last digits (0, 1, 8).
  • Sum of digits must be divisible by 9.
• What are rational numbers?


  Numbers that can be expressed as a fraction of two integers, where the denominator is not zero.