dec 21(part 1)(41:20)

Homework Review

• Start with reviewing homework completion
• Discuss discrepancies in notation and errors in solutions to questions
• Example Corrections:
  • 92 corrected to 9/2
  • 10² corrected to 100 (instead of 102)

Question Breakdown

• Discuss solvable questions individually or in total.

Question 1

• Problem: Odd page count between 253 and 523
• Approach:
  • Subtract page numbers to find total: 523 - 253 = 270
  • To find odd pages, calculate (n-1):
    • (523-253) - 1 * => yields total pages
  • Understand odd/even distribution based on the pages

Example Calculation

• Between 1 and 5, total pages = 5 - 1 = 3
• Odd numbers: 1, 3, 5 -> Total odd = 2
• For 269: totals are counted as odd and even based on page numbering.

Question 2

• Prime Factorization of 312:
  • Divisible by 2: 312 → 156 → 78 → 39 (not divisible by 2)
  • Next prime 3: 39/3 = 13 (prime)
  • Result: 2³ × 3¹ × 13¹ = 312.

Question 3

• Calculate GCD and LCM for 42, 72, 84:
  • Use factorization or shared divisors for numbers
  • GCD noted as 6, LCM calculated.

Question 4

• Identify perfect squares
  • Illustrate through examples such as:
    • 3/4 as perfect square (3/2)*(3/2)
  • Various products confirming perfect square status

Additional Perfect Square Considerations

• Memorization of numbers is crucial, knowing squares/cubes can expedite calculations.

Questions on Powers

• Reciprocal Definitions and Operations:
  • Definition:
    • Reciprocal of 3 is 1/3; key in calculations.
  • Simplifying powers using reciprocals.

Applying Orders of Magnitude

• Understanding scientific notation through approximations.
• Example: Estimate seconds in a year using:
  • Days x Hours x Minutes x Seconds -> Approximate values for manageable calculations.

Order of Operations (PEMDAS)

• Reminder of operational hierarchy for calculations:
  • Parentheses, Exponents, Multiplication/Division, Addition/Subtraction

Powers and Exponents

• Key properties include:
  • Same base multiplication: Add exponents
  • Division: Subtract exponents
  • Zero rule: Any number to the zero equals one.

Advanced Topics on Exponents

• Discuss fractional exponents and their interpretation (e.g., how to handle roots and powers during calculations).
• Clarification on roots - odd/even root handling.

Final Considerations during review session:

• Memorization Importance:
  • Perfect squares, cubes, and their fundamental properties is essential for efficiency in problems.
• Address any lingering questions or topics needing further clarity before transition to next lessons on real numbers and polynomials