Powers, Exponents & Geometry Basics

Powers And Exponents

• Definition: An exponent indicates how many times a number (base) is multiplied by itself.
• Notation: ( a^n ) where:
  • ( a ) = base
  • ( n ) = exponent
• Laws of Exponents:
  1. ( a^m \times a^n = a^{m+n} )
  2. ( \frac{a^m}{a^n} = a^{m-n} )
  3. ( (a^m)^n = a^{mn} )
  4. ( a^0 = 1 ) (where ( a \neq 0 ))
  5. ( a^{-n} = \frac{1}{a^n} )
• Special Cases:
  • ( 2^3 = 8 )
  • ( 10^2 = 100 )

Geometric Shapes

• Types of Shapes:
  • 2D Shapes: Circles, triangles, rectangles, squares, trapezoids.
  • 3D Shapes: Cubes, spheres, cylinders, cones, pyramids.
• Properties:
  • Triangles: Sum of angles = 180 degrees.
  • Quadrilaterals: Sum of angles = 360 degrees.
  • Circles: Radius (r), Diameter (d = 2r), Circumference (( C = \pi d )).

Area And Perimeter

• Area:

  • Rectangle: ( \text{Area} = \text{length} \times \text{width} )
  • Triangle: ( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} )
  • Circle: ( \text{Area} = \pi r^2 )

• Perimeter:

  • Rectangle: ( \text{Perimeter} = 2 \times (\text{length} + \text{width}) )
  • Triangle: ( \text{Perimeter} = \text{side1} + \text{side2} + \text{side3} )
  • Circle: ( \text{Circumference} = \pi d ) or ( 2\pi r )

Division Concepts

• Definition: Division is the process of finding out how many times one number (divisor) is contained in another (dividend).
• Notation: ( a \div b ) or ( \frac{a}{b} ) where:
  • ( a ) = dividend
  • ( b ) = divisor
• Key Concepts:
  • Quotient: Result of division.
  • Remainder: What is left over when division does not result in a whole number.
• Long Division: A method for dividing larger numbers step-by-step.
• Properties:
  • Division by 1: ( a \div 1 = a )
  • Division by itself: ( a \div a = 1 ) (where ( a \neq 0 ))

Powers And Exponents

• An exponent indicates the number of times a base is multiplied by itself.
• Notation for exponents is ( a^n ) where ( a ) is the base and ( n ) is the exponent.
• Laws of Exponents:
  • ( a^m \times a^n = a^{m+n} ): Multiplying like bases adds the exponents.
  • ( \frac{a^m}{a^n} = a^{m-n} ): Dividing like bases subtracts the exponents.
  • ( (a^m)^n = a^{mn} ): Raising a power to another power multiplies the exponents.
  • Any non-zero base raised to the power of zero equals one: ( a^0 = 1 ).
  • A negative exponent represents the reciprocal: ( a^{-n} = \frac{1}{a^n} ).
• Special cases include ( 2^3 = 8 ) and ( 10^2 = 100 ).

Geometric Shapes

• Geometric shapes are categorized into:
  • 2D Shapes: Include circles, triangles, rectangles, squares, and trapezoids.
  • 3D Shapes: Include cubes, spheres, cylinders, cones, and pyramids.
• Properties of shapes:
  • In triangles, the sum of interior angles equals 180 degrees.
  • For quadrilaterals, the sum of interior angles equals 360 degrees.
  • In circles, notable elements include radius (r), diameter (( d = 2r )), and circumference (( C = \pi d )).

Area And Perimeter

• Formulas for calculating area:
  • Rectangle: ( \text{Area} = \text{length} \times \text{width} ).
  • Triangle: ( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} ).
  • Circle: ( \text{Area} = \pi r^2 ).
• Formulas for calculating perimeter:
  • Rectangle: ( \text{Perimeter} = 2 \times (\text{length} + \text{width}) ).
  • Triangle: ( \text{Perimeter} = \text{side1} + \text{side2} + \text{side3} ).
  • Circle (Circumference): ( C = \pi d ) or ( 2\pi r ).

Division Concepts

• Division is the process of determining how many times a divisor can fit into a dividend.
• Notation for division includes ( a \div b ) or ( \frac{a}{b} ), where ( a ) is the dividend and ( b ) is the divisor.
• Key concepts in division:
  • Quotient: The result of the division operation.
  • Remainder: The leftover value when a dividend is not completely divisible by the divisor.
  • Long Division: A systematic method for dividing larger numbers.
• Key properties of division:
  • Dividing any number by 1 results in the same number: ( a \div 1 = a ).
  • Dividing a number by itself results in one, provided the number is not zero: ( a \div a = 1 ).

Geometric Shapes

• Basic Shapes include Circle, Triangle, Square, Rectangle, Pentagon, and Hexagon.
• Circle: Points are equidistant from a center; Area = πr² and Circumference = 2πr.
• Triangle: Angles sum to 180°; Area = 1/2 × base × height.
• Square: All sides are equal; Area = side² and Perimeter = 4 × side.
• Rectangle: Opposite sides are equal; Area = length × width and Perimeter = 2(length + width).
• Polygon: A closed shape with straight sides; Interior angle sum = (n-2) × 180°, where n is the number of sides.

Powers And Exponents

• Definition: Powers signify repeated multiplication.
• Notation: Expressed as a^n (a raised to power n).
• Base and Exponent: Base (a) is the number multiplied, and Exponent (n) shows how many times to multiply the base.
• Laws of Exponents:
  • a^m × a^n = a^(m+n)
  • a^m / a^n = a^(m-n)
  • (a^m)^n = a^(m×n)
  • a^0 = 1 (a ≠ 0)
  • a^(-n) = 1/a^n

Division Concepts

• Definition: Division determines how many times one number fits into another.
• Notation: a ÷ b = c, with a as dividend, b as divisor, and c as quotient.
• Key Terms:
  • Dividend: The number being divided.
  • Divisor: The number used for division.
  • Quotient: Result of the division operation.
• Properties:
  • Division by 1 results in the original number (a ÷ 1 = a).
  • Dividing a number by itself equals 1 (a ÷ a = 1, a ≠ 0).
  • Division by zero is undefined.

Area And Perimeter

• Area: Refers to the total space enclosed within a shape.
• Perimeter: Measures the complete distance around a shape.
• Formulas:
  • Rectangle: Area = length × width; Perimeter = 2(length + width).
  • Triangle: Area = 1/2 × base × height; Perimeter = sum of all sides.
  • Circle: Area = πr²; Perimeter (Circumference) = 2πr.

Volume And 3D Shapes

• Volume: Represents the total space occupied by a 3D shape.
• Common 3D Shapes and Formulas:
  • Cube: Volume = side³.
  • Rectangular Prism: Volume = length × width × height.
  • Cylinder: Volume = πr²h.
  • Sphere: Volume = (4/3)πr³.
  • Cone: Volume = (1/3)πr²h.
• Surface Area: Total area covering the surface of a 3D shape.
  • Cube: Surface Area = 6 × side².
  • Rectangular Prism: Surface Area = 2(lw + lh + wh).