Let's work through each part step by step.
### Part 1: Squaring a Two-Digit Number
1. Choose a two-digit number greater than 25:
- Let's choose [tex]\( x = 36 \)[/tex].
2. Rewrite 36 as a difference of two numbers:
- We can express [tex]\( 36 \)[/tex] as [tex]\( 40 - 4 \)[/tex].
- So, let [tex]\( y_1 = 40 \)[/tex] and [tex]\( y_2 = 4 \)[/tex].
3. Use the identity [tex]\((x - y)^2 = x^2 - 2xy + y^2\)[/tex]:
- We are squaring [tex]\( 36 \)[/tex] using the identity. Here [tex]\( x = y_1 \)[/tex] and [tex]\( y = y_2 \)[/tex].
- Calculate [tex]\( x^2 \)[/tex]:
[tex]\[
y_1^2 = 40^2 = 1600
\][/tex]
- Calculate [tex]\( -2xy \)[/tex]:
[tex]\[
-2 \cdot y_1 \cdot y_2 = -2 \cdot 40 \cdot 4 = -320
\][/tex]
- Calculate [tex]\( y^2 \)[/tex]:
[tex]\[
y_2^2 = 4^2 = 16
\][/tex]
- Combine the terms to find the square of 36:
[tex]\[
(36)^2 = 1600 - 320 + 16 = 1296
\][/tex]
### Part 2: Sum of the Cubes
1. Choose values for [tex]\( a \)[/tex] and [tex]\( b \)[/tex], each between 8 and 15:
- Let's choose [tex]\( a = 10 \)[/tex] and [tex]\( b = 12 \)[/tex].
2. Use the identity [tex]\( a^3 + b^3 = (a + b) (a^2 - ab + b^2) \)[/tex]:
- First, calculate [tex]\( a + b \)[/tex]:
[tex]\[
a + b = 10 + 12 = 22
\][/tex]
- Next, calculate [tex]\( a^2 \)[/tex]:
[tex]\[
a^2 = 10^2 = 100
\][/tex]
- Calculate [tex]\( ab \)[/tex]:
[tex]\[
ab = 10 \cdot 12 = 120
\][/tex]
- Calculate [tex]\( b^2 \)[/tex]:
[tex]\[
b^2 = 12^2 = 144
\][/tex]
- Combine the terms inside the parenthesis:
[tex]\[
a^2 - ab + b^2 = 100 - 120 + 144 = 124
\][/tex]
- Now, multiply [tex]\( (a + b) \)[/tex] by [tex]\( (a^2 - ab + b^2) \)[/tex]:
[tex]\[
a^3 + b^3 = 22 \times 124 = 2728
\][/tex]
By following these steps, we find that:
- The square of 36 is [tex]\( 1296 \)[/tex].
- The sum of the cubes of 10 and 12 is [tex]\( 2728 \)[/tex].
