Powers, Exponents & Geometry Basics

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Questions and Answers

An exponent indicates how many times a number (base) is multiplied by itself, represented as ______ where a is the base and n is the exponent.

a^n

The sum of angles in a triangle is ______ degrees.

180

The area of a rectangle is calculated using the formula: Area = ______ × width.

length

In division, the result of the operation is called the ______.

<p>quotient</p> Signup and view all the answers

For a circle, the relationship between radius and diameter is given by the formula d = ______.

<p>2r</p> Signup and view all the answers

The perimeter of a triangle is found by adding the lengths of its three sides, which can be expressed as ______.

<p>side1 + side2 + side3</p> Signup and view all the answers

According to the laws of exponents, a^m × a^n equals a raised to the power of ______.

<p>m+n</p> Signup and view all the answers

The area of a circle is calculated with the formula Area = ______ r^2.

<p>Ï€</p> Signup and view all the answers

The interior angle sum of an octagon is equal to 1080°.

<p>True (A)</p> Signup and view all the answers

The area of a triangle can be expressed as 2 × base × height.

<p>False (B)</p> Signup and view all the answers

If a is a non-zero number, then a^{-1} is equivalent to 1/a.

<p>True (A)</p> Signup and view all the answers

The perimeter of a square can be calculated as 6 × side.

<p>False (B)</p> Signup and view all the answers

The volume of a cuboid is calculated as length × width × height.

<p>True (A)</p> Signup and view all the answers

Circle's circumference is calculated using the formula πd.

<p>True (A)</p> Signup and view all the answers

In division, if the divisor is 0, the result is defined as a number.

<p>False (B)</p> Signup and view all the answers

The area of a square is calculated as the sum of its four sides.

<p>False (B)</p> Signup and view all the answers

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Study Notes

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 = Ï€r² and Circumference = 2Ï€r.
  • 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 = Ï€r²; Perimeter (Circumference) = 2Ï€r.

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 = Ï€r²h.
    • Sphere: Volume = (4/3)Ï€r³.
    • Cone: Volume = (1/3)Ï€r²h.
  • Surface Area: Total area covering the surface of a 3D shape.
    • Cube: Surface Area = 6 × side².
    • Rectangular Prism: Surface Area = 2(lw + lh + wh).

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