Question 175
Um cubo tem aresta de 3 cm. A diagonal do cubo, em cm, é
(A) $3\sqrt{2}$ (B) $3\sqrt{3}$ (C) $6\sqrt{2}$ (D) $6\sqrt{3}$ (E) $9\sqrt{2}$

Among other research objects, Allometry studies the relationship between measurements of different parts of the human body. For example, according to Allometry, the area A of the body surface of a person relates to their mass $m$ by the formula $A = k \cdot \mathrm{~m}^{\frac{2}{3}}$, where $k$ is a positive constant.
If during the period from childhood to adulthood of an individual their mass is multiplied by 8, by how much will the area of body surface be multiplied?
(A) $\sqrt[3]{16}$
(B) 4
(C) $\sqrt{24}$
(D) 8
(E) 64

QUESTION 170
The value of $2^{10}$ is
(A) 512
(B) 1024
(C) 2048
(D) 4096
(E) 8192

QUESTION 175
The value of $\sqrt{144} + \sqrt{25}$ is
(A) 13
(B) 15
(C) 17
(D) 19
(E) 21

The value of $2^{10}$ is:
(A) 512
(B) 1024
(C) 2048
(D) 4096
(E) 8192

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

What is the value of $3 ^ { \frac { 2 } { 3 } } \times 9 ^ { \frac { 3 } { 2 } } \div 27 ^ { \frac { 8 } { 9 } }$? [2 points]
(1) 1
(2) $\sqrt { 3 }$
(3) 3
(4) $3 \sqrt { 3 }$
(5) 9

What is the value of $3 ^ { \frac { 2 } { 3 } } \times 9 ^ { \frac { 3 } { 2 } } \div 27 ^ { \frac { 8 } { 9 } }$? [2 points]
(1) 1
(2) $\sqrt { 3 }$
(3) 3
(4) $3 \sqrt { 3 }$
(5) 9

What is the value of $5 ^ { \frac { 2 } { 3 } } \times 25 ^ { - \frac { 5 } { 6 } }$? [2 points]
(1) $\frac { 1 } { 25 }$
(2) $\frac { 1 } { 5 }$
(3) 1
(4) 5
(5) 25

What is the value of $5 ^ { \frac { 2 } { 3 } } \times 25 ^ { - \frac { 5 } { 6 } }$? [2 points]
(1) $\frac { 1 } { 25 }$
(2) $\frac { 1 } { 5 }$
(3) 1
(4) 5
(5) 25

When $a = \sqrt { 2 } , b ^ { 3 } = \sqrt { 3 }$, what is the value of $( a b ) ^ { 2 }$? (Here, $b$ is a real number.) [3 points]
(1) $2 \cdot 3 ^ { \frac { 1 } { 3 } }$
(2) $2 \cdot 3 ^ { \frac { 2 } { 3 } }$
(3) $2 ^ { \frac { 1 } { 2 } } \cdot 3 ^ { \frac { 1 } { 3 } }$
(4) $3 \cdot 2 ^ { \frac { 1 } { 3 } }$
(5) $3 \cdot 2 ^ { \frac { 2 } { 3 } }$

What is the value of $9 ^ { \frac { 3 } { 2 } } \times 27 ^ { - \frac { 2 } { 3 } }$? [2 points]
(1) $\frac { 1 } { 3 }$
(2) 1
(3) $\sqrt { 3 }$
(4) 3
(5) $3 \sqrt { 3 }$

What is the value of $9 ^ { \frac { 3 } { 2 } } \times 27 ^ { - \frac { 2 } { 3 } }$? [2 points]
(1) $\frac { 1 } { 3 }$
(2) 1
(3) $\sqrt { 3 }$
(4) 3
(5) $3 \sqrt { 3 }$

What is the value of $27 ^ { \frac { 1 } { 3 } } + \log _ { 2 } 4$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

What is the value of $27 ^ { \frac { 1 } { 3 } } + \log _ { 2 } 4$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For natural numbers $n$ with $2 \leq n \leq 100$, find the number of values of $n$ such that $\left( \sqrt[3]{3^5} \right)^{\frac{1}{2}}$ is the $n$-th root of some natural number. [4 points]

What is the value of $8 ^ { \frac { 2 } { 3 } } \times 9 ^ { \frac { 1 } { 2 } }$? [2 points]
(1) 4
(2) 6
(3) 8
(4) 10
(5) 12

What is the value of $5 \times 8 ^ { \frac { 1 } { 3 } }$? [2 points]
(1) 10
(2) 15
(3) 20
(4) 25
(5) 30

What is the sum of all natural numbers $x$ that satisfy the exponential inequality $\left( \frac { 1 } { 5 } \right) ^ { 1 - 2 x } \leq 5 ^ { x + 4 }$? [4 points]
(1) 11
(2) 12
(3) 13
(4) 14
(5) 15

What is the value of $8 ^ { \frac { 1 } { 3 } } + 27 ^ { \frac { 2 } { 3 } }$? [2 points]
(1) 8
(2) 9
(3) 10
(4) 11
(5) 12

What is the value of $8 \times 2 ^ { - 2 }$? [2 points]
(1) 1
(2) 2
(3) 4
(4) 8
(5) 16

Find the value of $2 \times 16 ^ { \frac { 1 } { 4 } }$. [2 points]
(1) 2
(2) 4
(3) 6
(4) 8
(5) 10

What is the value of $2 ^ { - 1 } \times 16 ^ { \frac { 1 } { 2 } }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

What is the value of $16 \times 2 ^ { - 3 }$? [2 points]
(1) 1
(2) 2
(3) 4
(4) 8
(5) 16

What is the value of $3 ^ { 0 } \times 8 ^ { \frac { 2 } { 3 } }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5