Powers, Exponents & Geometry Basics

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 )

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:
- 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 )

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 ))

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 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 )).

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 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 ).

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.

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

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: 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: 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).