Math Class: Rational Numbers and Exponents

Study Notes

Rational Numbers: A rational number is a number that can be expressed as the ratio of two integers, i.e., p/q, where p and q are integers and q ≠ 0.
- Examples: 3/4, 22/7, 1/2
- Properties:
  - Commutative property: a/b = b/a
  - Associative property: (a/b) × (c/d) = a/c × b/d
  - Distributive property: a/b + c/d = (a+c)/(b+d)
Exponents and Powers: An exponent is a small number that is raised to a power to indicate the number of times a base number should be multiplied by itself.
- Laws of Exponents:
  - Product of powers: a^m × a^n = a^(m+n)
  - Power of a power: (a^m)^n = a^(m×n)
  - Quotient of powers: a^m ÷ a^n = a^(m-n)
Squares and Square Roots: A square of a number is the result of multiplying the number by itself.
- Properties:
  - Square of a sum: (a+b)^2 = a^2 + 2ab + b^2
  - Square of a difference: (a-b)^2 = a^2 - 2ab + b^2
  - Square root of a number: √a = b, where b^2 = a

Linear Equations: An equation in which the highest power of the variable is 1.
- General form: ax + by = c, where a, b, and c are constants
- Solution: x = (c - by) / a
Quadratic Equations: An equation in which the highest power of the variable is 2.
- General form: ax^2 + bx + c = 0, where a, b, and c are constants
- Solution: x = (-b ± √(b^2 - 4ac)) / 2a

Understanding Shapes: A shape is a self-contained area with a fixed boundary.
- Types of shapes:
  - 2D shapes: triangles, quadrilaterals, polygons, circles
  - 3D shapes: cubes, cuboids, spheres, cones
Properties of Shapes: Properties of shapes include:
- Angles: acute, obtuse, right, straight, reflex
- Sides: number of sides, length of sides
- Vertices: number of vertices, properties of vertices

Perimeter and Area: Perimeter is the distance around a shape, while area is the amount of space inside a shape.
- Formulas:
  - Perimeter of a rectangle: 2(l + b)
  - Area of a rectangle: l × b
  - Perimeter of a triangle: a + b + c
  - Area of a triangle: (b × h) / 2
Volume and Capacity: Volume is the amount of space inside a 3D shape, while capacity is the amount of liquid a container can hold.
- Formulas:
  - Volume of a cube: s^3
  - Volume of a cuboid: l × b × h
  - Capacity of a container: volume of liquid / volume of container

Rational Numbers: A number that can be expressed as the ratio of two integers, p/q, where p and q are integers and q ≠ 0.
Examples of rational numbers: 3/4, 22/7, 1/2
Properties of rational numbers:
- Commutative property: a/b = b/a
- Associative property: (a/b) × (c/d) = a/c × b/d
- Distributive property: a/b + c/d = (a+c)/(b+d)

An exponent is a small number that is raised to a power to indicate the number of times a base number should be multiplied by itself.
Laws of Exponents:
- Product of powers: a^m × a^n = a^(m+n)
- Power of a power: (a^m)^n = a^(m×n)
- Quotient of powers: a^m ÷ a^n = a^(m-n)

A square of a number is the result of multiplying the number by itself.
Properties of squares:
- Square of a sum: (a+b)^2 = a^2 + 2ab + b^2
- Square of a difference: (a-b)^2 = a^2 - 2ab + b^2
Square root of a number: √a = b, where b^2 = a

Linear Equations: An equation in which the highest power of the variable is 1.
General form of linear equations: ax + by = c, where a, b, and c are constants
Solution of linear equations: x = (c - by) / a

Quadratic Equations: An equation in which the highest power of the variable is 2.
General form of quadratic equations: ax^2 + bx + c = 0, where a, b, and c are constants
Solution of quadratic equations: x = (-b ± √(b^2 - 4ac)) / 2a

Understanding Shapes: A shape is a self-contained area with a fixed boundary.
Types of shapes:
- 2D shapes: triangles, quadrilaterals, polygons, circles
- 3D shapes: cubes, cuboids, spheres, cones
Properties of Shapes: Properties of shapes include:
- Angles: acute, obtuse, right, straight, reflex
- Sides: number of sides, length of sides
- Vertices: number of vertices, properties of vertices

Perimeter and Area: Perimeter is the distance around a shape, while area is the amount of space inside a shape.
Formulas for perimeter and area:
- Perimeter of a rectangle: 2(l + b)
- Area of a rectangle: l × b
- Perimeter of a triangle: a + b + c
- Area of a triangle: (b × h) / 2

Volume and Capacity: Volume is the amount of space inside a 3D shape, while capacity is the amount of liquid a container can hold.
Formulas for volume and capacity:
- Volume of a cube: s^3
- Volume of a cuboid: l × b × h
- Capacity of a container: volume of liquid / volume of container