Math: BEDMAS, Exponents & Place Value

ORDER OF OPERATIONS — BEDMAS

BEDMAS tells you the order to do math operations:

  B = Brackets (do what's inside brackets first)
  E = Exponents (powers like 2³ or 10²)
  D = Division  }  do these LEFT TO RIGHT
  M = Multiplication  }  (whichever comes first)
  A = Addition  }  do these LEFT TO RIGHT
  S = Subtraction  }  (whichever comes first)

  CRITICAL: Division and Multiplication have EQUAL priority.
  Addition and Subtraction have EQUAL priority.
  Within each pair, go left to right.

BEDMAS EXAMPLES

  Example 1: 2 + 2 × 4 + 2
    Multiply first: 2 × 4 = 8
    Then left to right: 2 + 8 + 2 = 12
    WRONG approach: (2+2) × (4+2) = 24 ← NO!

  Example 2: 3 × 2 + 4 ÷ 4
    Multiply: 3 × 2 = 6
    Divide: 4 ÷ 4 = 1
    Add: 6 + 1 = 7

  Example 3: 48 + 23 × 9 - (65 - 3)
    Brackets first: 65 - 3 = 62
    Multiply: 23 × 9 = 207
    Left to right: 48 + 207 - 62 = 193

  Example 4: (14.8 + 11.6)² + 20.7 ÷ 0.9
    Brackets: 14.8 + 11.6 = 26.4
    Exponent: 26.4² = 696.96
    Divide: 20.7 ÷ 0.9 = 23
    Add: 696.96 + 23 = 719.96

  THE MOST COMMON MISTAKE: Adding before multiplying.
  "2 + 3 × 4" is 14, not 20.

EXPONENT RULES

An exponent tells you how many times to multiply a number by itself.

  3⁴ = 3 × 3 × 3 × 3 = 81

THE THREE RULES YOU NEED:

  1. PRODUCT RULE: Same base, multiplying → ADD exponents
     a^m × a^n = a^(m+n)

     Example: 3¹¹ × 3² = 3^(11+2) = 3¹³

  2. QUOTIENT RULE: Same base, dividing → SUBTRACT exponents
     a^m ÷ a^n = a^(m-n)

     Example: a¹² ÷ a⁴ = a^(12-4) = a⁸

  3. NEGATIVE EXPONENTS: Work the same way
     3⁻⁶ × 3² = 3^(-6+2) = 3⁻⁴

  COMMON MISTAKES:
  - Multiplying the exponents instead of adding (3¹¹ × 3² ≠ 3²²)
  - Multiplying the bases (3¹¹ × 3² ≠ 9¹³)
  - The base stays the same. Only the exponents change.

SOLVING EXPONENT EQUATIONS

  Example: (a¹²)(aˣ) / a⁴ = a²

  Step 1: Combine the left side using exponent rules.
    a^(12 + x) / a⁴ = a^(12 + x - 4) = a^(8 + x)

  Step 2: Set the exponents equal.
    8 + x = 2

  Step 3: Solve.
    x = 2 - 8 = -6

PLACE VALUE

Every digit in a number has a value based on its position.

  Millions | Hundred-thousands | Ten-thousands | Thousands | Hundreds | Tens | Ones
  10^6     | 10^5              | 10^4          | 10^3      | 10^2     | 10^1 | 10^0

  Example: "19 thousands + 7 tens"
    19 × 1,000 = 19,000
    7 × 10 = 70
    Total = 19,070

  To find the value of a specific digit: multiply the digit by its
  place value.

EXPANDED FORM

Writing a number as a sum of each digit times its place value.

  4,937 = 4 × 1,000 + 9 × 100 + 3 × 10 + 7 × 1

  READING expanded form back to a number:

  Example: (3 × 1,000,000) + (2 × 10,000) + (7 × 1,000) + (9 × 100) + (6 × 10) + (8 × 1)
    = 3,000,000 + 20,000 + 7,000 + 900 + 60 + 8
    = 3,027,968

  WATCH OUT FOR MISSING PLACES: If there's no 100,000 term in the
  expanded form, that place value is 0. In the example above, there's
  no 10^5 term, so the number is 3,027,968 not 3,27,968.

  Example: (7 × 1,000,000) + (3 × 10,000) + (6 × 100) + (2 × 10) + (9 × 1)
    = 7,000,000 + 30,000 + 600 + 20 + 9
    = 7,030,629
    (Missing: hundred-thousands AND thousands — both are 0)

KEY FORMULAS

  BEDMAS: Brackets → Exponents → Division/Multiplication → Addition/Subtraction
  Product rule: a^m × a^n = a^(m+n)
  Quotient rule: a^m ÷ a^n = a^(m-n)
  Negative exponents: same rules, just keep the negative signs
  Expanded form: each digit × its place value, added together
  Missing places in expanded form = zeros in the final number