Maths

• What is the general form of the difference of squares?

  $x^2 - y^2$
• What are the characteristics of the difference of squares pattern?

  It consists of two squared terms connected by a minus sign.
• What is the expression we want to show is the product of two consecutive odd numbers?

  $2^{40} - 1$
• What algebraic identity is used to expand $2^{40} - 1$?


  - $a^n - 1 =$ \left(a^{n/2} - 1\right)\left(a^{n/2} + 1\right) - Applicable when n is even
• How can we rewrite $2^{40} - 1$ using the algebraic identity?


  $2^{40} - 1 =$$(2^{20} - 1)(2^{20} +$$1)$
• What is the first step in simplifying $2^{40} - 1$?


  1. Recognize $2^{40}$ as $(2^{20})^2$ 2. Apply the identity: $(x^2 - 1 = (x-1)(x+1))$
• What is the result of $2^{10}$?


  $1024$
• What does the expression $(2^{20} - 1)(2^{20} +$$1)$ represent after factoring?


  $2^{40} - 1 =$$(2^{10} - 1)(2^{10} +$$1)(2^{20} +$$1)$
• What are the two main factors of $2^{40} - 1$ after factoring?


  - Factor 1: $2^{10} - 1$ - Factor 2: $(2^{10} +$$1)(2^{20} +$$1)$
• How do we verify that $1023$ and $1074790401$ are consecutive odd numbers?


  1. Check if both numbers are odd: - $1023 \div 2 =$$511 \text{ remainder } 1$ - $1074790401 \div 2 =$$537395200 \text{ remainder } 1$ 2. Calculate the difference: - $1074790401 - 1023 =$$1074789378$ 3. Check if the difference is divisible by 2: - $1074789378 \div 2 =$$537394689$ (no remainder)
• What confirms that two numbers are consecutive odd numbers?

  The difference between them is divisible by 2 with no whole numbers in between.
• Which pair of numbers are consecutive odd numbers?

  15 and 17
• What defines an isosceles triangle?

  An isosceles triangle has two sides of equal length.