$$\frac { \frac { 4 } { 3 } + \frac { 3 } { 4 } } { \frac { 2 } { 3 } - \frac { 1 } { 4 } }$$
What is the result of this operation?
A) 5
B) 10
C) 15
D) 20
E) 25

For integers a and b
$$\frac { 6 ^ { a ^ { 2 } + b ^ { 2 } } } { 9 ^ { a b } } = 96$$
Given that, what is the product $\mathbf { a } \cdot \mathbf { b }$?
A) 1 B) 2 C) 3 D) 4 E) 6

$$\frac { 6 ^ { 4 } - 4 ^ { 4 } } { 5 \cdot 2 ^ { 4 } }$$
What is the result of this operation?
A) 9
B) 12
C) 13
D) 14
E) 16

$$\frac { 4 } { 9 - \frac { 49 } { 9 } } - \frac { 1 } { 8 }$$
What is the result of this operation?
A) 1
B) 2
C) 3
D) $\frac { 1 } { 2 }$
E) $\frac { 1 } { 4 }$

$$\frac { 6 ^ { -8 } \cdot 9 ^ { 4 } } { 4 ^ { -6 } }$$
What is the result of this operation?
A) 8
B) 9
C) 12
D) 16
E) 18

$$\frac { \sqrt { 48 } } { \frac { 1 } { \sqrt { 3 } } + \frac { 1 } { \sqrt { 27 } } }$$
What is the result of this operation?
A) 3
B) 5
C) 8
D) 9
E) 12

$$\frac { \sqrt { 12 } } { \sqrt { 27 } + \frac { 1 } { \sqrt { 3 } } }$$
What is the result of this operation?
A) $\frac { 2 } { 3 }$
B) $\frac { 3 } { 5 }$
C) $\frac { 1 } { 2 }$
D) $\sqrt { 3 }$
E) $\sqrt { 6 }$

$$\begin{aligned} & a = \sqrt { 2 } + \sqrt { 45 } \\ & b = \sqrt { 5 } + \sqrt { 18 } \\ & c = \sqrt { 8 } + \sqrt { 20 } \end{aligned}$$
Given this, which of the following orderings is correct?
A) a $<$ b $<$ c
B) b $<$ a $<$ c
C) c $<$ b $<$ a
D) b $<$ c $<$ a
E) c $<$ a $<$ b

For integers a and b
$$16 ^ { a } \cdot 9 ^ { a } = 6 ^ { b } \cdot 8 ^ { 2 }$$
Given this equality, what is the sum $\mathbf { a } + \mathbf { b }$?
A) 6
B) 9
C) 12
D) 15
E) 20

$$\frac { 1 } { \sqrt { 2 x } } + \frac { 4 } { \sqrt { 8 x } } = 6$$
Given this, what is $x$?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 6 }$
D) $\frac { 1 } { 8 }$
E) $\frac { 1 } { 12 }$

$$\sqrt [ 3 ] { \frac { 32 } { \sqrt { 8 } - \sqrt { 2 } } }$$
What is the result of this operation?
A) $\sqrt { 2 }$ B) $2 \sqrt { 2 }$ C) $\sqrt [ 3 ] { 2 }$ D) 2 E) 4

$\frac { 1 } { \sqrt { \mathrm{a} } } - \frac { 2 } { \sqrt { 9 \mathrm{a} } } = 1$\ Given this, what is a?\ A) $\frac { 1 } { 3 }$\ B) $\frac { 2 } { 3 }$\ C) $\frac { 1 } { 4 }$\ D) $\frac { 1 } { 9 }$\ E) $\frac { 4 } { 9 }$

For integers $x$ and $y$
$$9 ^ { x } - 3 ^ { 2 x - 2 } = 2 ^ { y } \cdot 3 ^ { 6 }$$
the equality is satisfied. Accordingly, what is the sum $x + y$?
A) 3
B) 4
C) 5
D) 6
E) 7

When the numbers $\sqrt{5}, \sqrt{8}, \sqrt{12}, \sqrt{18}, \sqrt{20}$ and $\sqrt{27}$ are placed in the boxes below, with each box containing a different number, A, B, and C become whole numbers.
Accordingly, what is the sum $\mathrm{A} + \mathrm{B} + \mathrm{C}$?
A) 40
B) 44
C) 48
D) 52
E) 56

For distinct natural numbers $a$, $b$, and $c$,
$$\frac { 6 ^ { a } \cdot 15 ^ { b } } { 9 ^ { b } \cdot 10 ^ { c } }$$
is equal to an integer. Accordingly, which of the following orderings is correct?
A) $a < b < c$
B) $b < a < c$
C) $b < c < a$
D) $c < a < b$
E) $c < b < a$

Mert, who performs operations with radical numbers, instead of multiplying the number $\sqrt{10} + \sqrt{6}$ by its conjugate $\sqrt{10} - \sqrt{6}$, mistakenly divided it.
Accordingly, how much greater is the number Mert found than the number he should have found?
A) $\sqrt{12}$ B) $\sqrt{15}$ C) $\sqrt{18}$ D) $\sqrt{20}$ E) $\sqrt{30}$

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) $\sqrt{80}$

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