Mathematics: Squares, Cubes, and Roots

Squares

• Definition: A square of a number is the result of multiplying that number by itself.
• Notation: ( n^2 )
• Examples:
  • ( 2^2 = 4 )
  • ( 3^2 = 9 )
• Properties:
  • Always non-negative (square of any real number).
  • ( n^2 ) grows faster than linear functions.

Cubes

• Definition: A cube of a number is the result of multiplying that number by itself twice.
• Notation: ( n^3 )
• Examples:
  • ( 2^3 = 8 )
  • ( 3^3 = 27 )
• Properties:
  • Can be positive or negative (odd power preserves the sign).
  • ( n^3 ) grows faster than squares for ( |n| > 1 ).

Roots

• Square Root:

  • Definition: A number that produces a specified quantity when multiplied by itself.
  • Notation: ( \sqrt{n} )
  • Examples:
    • ( \sqrt{4} = 2 )
    • ( \sqrt{9} = 3 )

• Cube Root:

  • Definition: A number that produces a specified quantity when used in a multiplication three times.
  • Notation: ( \sqrt[3]{n} )
  • Examples:
    • ( \sqrt[3]{8} = 2 )
    • ( \sqrt[3]{27} = 3 )

Division

• Definition: The operation of determining how many times one number is contained within another.
• Notation: ( \frac{a}{b} ) or ( a \div b )
• Properties:
  • Division by zero is undefined.
  • Can be expressed in terms of multiplication: ( a \div b = a \times \frac{1}{b} ).
  • The quotient and remainder are concepts relevant in integer division.

Multiplication

• Definition: The operation of scaling one number by another.
• Notation: ( a \times b ) or ( ab )
• Properties:
  • Commutative: ( a \times b = b \times a )
  • Associative: ( a \times (b \times c) = (a \times b) \times c )
  • Distributive: ( a \times (b + c) = (a \times b) + (a \times c) )
• Identity Element: The number 1 (e.g., ( a \times 1 = a )).

Squares

• The square operation involves multiplying a number by itself.
• It is denoted by ( n^2 ), where ( n ) is the number.
• Squares are always positive or zero.
• (n^2 ) increases faster than linear functions.
• Examples include: ( 2^2 = 4 ) and ( 3^2 = 9 )

Cubes

• The cube operation involves multiplying a number by itself twice.
• It is denoted by ( n^3 ), where ( n ) is the number.
• Cubes can be positive or negative as an odd number of multiplications preserves the sign.
• ( n^3 ) increases faster than squares when the absolute value of ( n ) is greater than 1.
• Examples include: ( 2^3 = 8 ) and ( 3^3 = 27 )

Roots

• The square root of a number is the number that results in the original number when multiplied by itself.

• It is denoted by ( \sqrt{n} ), where ( n ) is the number.

• Examples include: ( \sqrt{4} = 2 ) and ( \sqrt{9} = 3 )

• The cube root of a number is the number that results in the original number when multiplied by itself three times.

• Similarly, it is denoted by ( \sqrt{n} ), where ( n ) is the number.

• Examples include: ( \sqrt{8} = 2 ) and ( \sqrt{27} = 3 )

Division

• The operation of division determines how many times one number is contained within another.
• It is denoted by ( \frac{a}{b} ) or ( a \div b ), where ( a ) is the dividend, and ( b ) is the divisor.
• Dividing by zero is undefined.
• Division can be represented in terms of multiplication: ( a \div b = a \times \frac{1}{b} )
• Concepts of the quotient (result) and remainder are important in integer division.

Multiplication

• Multiplication is the operation of scaling one number by another.
• It is denoted by ( a \times b ) or ( ab ), where ( a ) and ( b ) are the multiplicands.
• Multiplication is commutative (order doesn't matter) and associative (grouping doesn't matter).
• The distributive property allows distributing multiplication over addition.
• The identity element for multiplication is 1, as ( a \times 1 = a ).