Statements
(17) $4 < \sqrt { 5 + 5 \sqrt { 5 } }$. (18) $\log _ { 2 } 11 < \frac { 1 + \log _ { 2 } 61 } { 2 }$. (19) $( 2023 ) ^ { 2023 } < ( 2023 ! ) ^ { 2 }$. (20) $92 ^ { 100 } + 93 ^ { 100 } < 94 ^ { 100 }$.

Among the three numbers $2 ^ { - 3 }$, $3 ^ { \frac { 1 } { 2 } }$, $\log _ { 2 } 5$, the largest is

Answer the following questions.
(1) The positive integers $m$ and $n$ which simultaneously satisfy the following two inequalities
$$\frac { m } { 3 } < \sqrt { 3 } < \frac { n } { 4 } , \quad \frac { n } { 3 } < \sqrt { 6 } < \frac { m } { 2 }$$
are
$$m = \mathbf { A } , \quad n = \mathbf { B } .$$
(2) Using the results of (1), let us compare the sizes of numbers (1) $\sim$ (5).
(1) $( \sqrt { ( - 3 ) ( - 4 ) } ) ^ { 3 }$
(2) $6 \sqrt { ( - 2 ) ^ { 3 } ( - 3 ) }$
(3) $\sqrt { \left\{ ( - 4 ) ( - 3 ) ^ { 2 } \right\} ^ { 2 } }$
(4) $( - 1 ) ^ { 3 } \sqrt { \left\{ ( - 2 ) ^ { 5 } \right\} ^ { 2 } }$
(5) $\left( \frac { 5 \sqrt { 3 } } { 1 - \sqrt { 6 } } \right) ^ { 2 }$
When the denominator of (5) is rationalized, we have
$$\left( \frac { 5 \sqrt { 3 } } { 1 - \sqrt { 6 } } \right) ^ { 2 } = \mathbf { C D } + \mathbf { E } \sqrt { \mathbf { F } }$$
Of the five numbers, there are $\mathbf { G }$ number(s) greater than 35 and $\mathbf { H }$ negative number(s).
When we arrange the five numbers in the ascending order of their size using the numbers (1) $\sim$ (5), we have
$$\mathbf { I } < \mathbf { J } < \mathbf { K } < \mathbf { L } < \mathbf { M } .$$

Given $a = 6$, $b = \frac{20}{3}$, $c = 2\sqrt{10}$, and $d$, where $d$ is a rational number. These four numbers are marked on a number line as $A(a)$, $B(b)$, $C(c)$, and $D(d)$. Select the correct options.
(1) $a + b + c + d$ must be a rational number (2) $abcd$ must be an irrational number (3) Point $D$ could possibly be at a distance of $2\sqrt{10} + 6$ from point $C$ (4) The midpoint of points $A$ and $B$ is to the right of point $C$ (5) Among all points on the number line at a distance less than 8 from point $B$, there are 14 positive integers and 1 negative integer

$x = \sqrt[3]{4}$
$$y = \sqrt[4]{8}$$ $$z = \sqrt[5]{16}$$
Given that, which of the following orderings is correct?
A) $x < y < z$ B) $x < z < y$ C) $y < x < z$ D) $z < x < y$ E) $z < y < x$

$$\begin{aligned} & a = \sqrt { 2 } + \sqrt { 45 } \\ & b = \sqrt { 5 } + \sqrt { 18 } \\ & c = \sqrt { 8 } + \sqrt { 20 } \end{aligned}$$
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