For integers $x$ and $y$
$$2^{3x-1} - 8^{x-1} = 3^{y+3} \cdot 4^{x+1}$$
Given this equality.
Accordingly, what is the product $\mathbf{x} \cdot \mathbf{y}$?
A) $-10$ B) $-6$ C) $-2$ D) 4 E) 8

Let $A$ and $B$ be natural numbers. A square with side length $A\sqrt{B}$ units has an area of 720 square units.
Accordingly, which of the following cannot be the sum $A + B$?
A) 26 B) 49 C) 83 D) 127 E) 182

A note that Elif read in her mathematics book states that $\sqrt{a} + \sqrt{b} = \sqrt{c}$ for some values. Because water dripped on her book, Elif could not read the number $b$ in the example.
Accordingly, which of the following cannot be the number $b$?
A) $\sqrt{5}$
B) $\sqrt{20}$
C) $\sqrt{45}$
D) $\sqrt{60}$
E) $\sqrt{80}$

A tablet screen; when "large icons" is selected in the display settings is divided into 3 rows and 4 columns into identical compartments, and when "small icons" is selected, it is divided into 5 rows and 6 columns into identical compartments as shown in Figure 2. In both cases, at most one application icon is placed in each compartment.
When "large icons" is selected in the display settings, it is observed that the icons of all applications on the tablet are located in $\frac{2}{3}$ of the compartments on the screen.
If the display settings are changed to "small icons", in what fraction of the compartments on the screen will the icons of all applications on the tablet be located?
A) $\frac{1}{5}$ B) $\frac{1}{6}$ C) $\frac{3}{10}$ D) $\frac{4}{15}$ E) $\frac{7}{30}$

While teaching the topic of exponents, Teacher Kerem stated that the expression $a^{b^{c}}$ cannot be written without parentheses in this way, because the expressions $a^{\left(b^{c}\right)}$ and $\left(a^{b}\right)^{c}$ can have different values, and explained this situation with an example.
Accordingly, which of the following could be the example given by Teacher Kerem?
A) $a = 1, b = 2, c = 3$ B) $a = 2, \quad b = 1, \quad c = 3$ C) $a = 2, \quad b = 2, \quad c = 2$ D) $a = 3, \quad b = 0, \quad c = 3$ E) $a = 3, \quad b = 2, \quad c = 1$

In a page of a mathematics textbook shown partially below, the result of the 1st operation is 12 more than the result of the 2nd operation.
$a = 2 \quad b = $
For the real numbers $a$ and $b$ given above, find the result of the following operations.
1. operation: $a\sqrt{b} + \sqrt{b} = $ 2. operation: $a\sqrt{b} - \sqrt{b} = $ 3. operation: $a\sqrt{b} \times \sqrt{b} = $ 4. operation: $a\sqrt{b} \div \sqrt{b} = $
Accordingly, the result of the 3rd operation is equal to how many times the result of the 4th operation?
A) 9 B) 16 C) 24 D) 30 E) 36

A mathematics teacher asked students to find the decimal representation of the number $\frac{13}{20}$. Sude, who found this representation correctly, accidentally swapped the tenths and hundredths digits when writing the number she found on the board.
Accordingly, the number Sude wrote on the board is equal to which of the following?
A) $\frac{5}{8}$ B) $\frac{9}{20}$ C) $\frac{14}{25}$ D) $\frac{23}{40}$ E) $\frac{27}{50}$