If a positive integer $N$ is entered into a calculator, and then the ``$\sqrt{ }$'' key (taking the positive square root) is pressed 3 times consecutively, the display shows the answer as 2. Then $N$ equals which of the following options?
(1) $2^{3}$
(2) $2^{4}$
(3) $2^{6}$
(4) $2^{8}$
(5) $2^{12}$

$( 16 ) ^ { 3 n } = 8 ^ { 5 }$
Given this, what is n?
A) $\frac { 3 } { 2 }$
B) $\frac { 4 } { 3 }$
C) $\frac { 3 } { 5 }$
D) $\frac { 5 } { 4 }$
E) $\frac { 5 } { 6 }$

$$\left( 1 - 3 ^ { - 1 } + a ^ { - 1 } \right) ^ { - 3 } = 8$$
Given this, what is a?
A) $- 6$
B) $- 4$
C) $\frac { - 2 } { 3 }$
D) $\frac { 3 } { 4 }$
E) $\frac { 1 } { 6 }$

$$\frac { 2 ^ { - 2 } } { 4 ^ { - 1 } + \frac { 1 } { m ^ { - 1 } } } = 13 ^ { - 1 }$$
Given this, what is m?
A) 3
B) 4
C) 5
D) 6
E) 7

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

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