The problem consists of two questions. Question 1 asks to evaluate several expressions involving exponents. Question 2 consists of two parts: Part a asks to find the values of some expressions involving exponents, given the value of $x$, and Part b asks to solve an equation.

1. Problem Description

The problem consists of two questions. Question 1 asks to evaluate several expressions involving exponents. Question 2 consists of two parts: Part a asks to find the values of some expressions involving exponents, given the value of $x$, and Part b asks to solve an equation.

2. Solution Steps

Question 1:

i. $(-0.5)^3 = (-1/2)^3 = (-1)^3 / 2^3 = -1/8 = -0.125$

ii. $27^{2/3} + 1 = (27^{1/3})^2 + 1 = (3)^2 + 1 = 9 + 1 = 10$

We use the formula:

$(a^m)^n = a^{mn}$

iii. $(\frac{25}{100})^{-1/2} + (\frac{3}{2})^0 = (\frac{100}{25})^{1/2} + 1 = (4)^{1/2} + 1 = 2 + 1 = 3$

We use the formulas:

$a^{-n} = \frac{1}{a^n}$

$a^0 = 1$

iv. $3^0 \times 3^3 + 3 = 1 \times 3^3 + 3 = 1 \times 27 + 3 = 27 + 3 = 30$

We use the formula:

$a^0 = 1$

v. $25^2 \div 5^5 = (5^2)^2 \div 5^5 = 5^4 \div 5^5 = \frac{5^4}{5^5} = \frac{1}{5}$

We use the formula:

$\frac{a^m}{a^n} = a^{m-n}$

vi. $2^3 \times 2^{-1} + x^0 = 8 \times \frac{1}{2} + 1 = 4 + 1 = 5$, assuming $x \neq 0$.

We use the formulas:

$a^{-n} = \frac{1}{a^n}$

$a^0 = 1$

Question 2:

a. Given $x = \frac{25}{16}$

i. $x^0 = (\frac{25}{16})^0 = 1$

ii. $x^{1/2} = (\frac{25}{16})^{1/2} = \sqrt{\frac{25}{16}} = \frac{\sqrt{25}}{\sqrt{16}} = \frac{5}{4}$

iii. $x^{-3/2} = (\frac{25}{16})^{-3/2} = (\frac{16}{25})^{3/2} = ((\frac{16}{25})^{1/2})^3 = (\frac{4}{5})^3 = \frac{4^3}{5^3} = \frac{64}{125}$

b. $\frac{6}{x+2} = \frac{2}{3}$

Cross-multiplying, we get:

$6 \times 3 = 2 \times (x+2)$

$18 = 2x + 4$

$2x = 18 - 4$

$2x = 14$

$x = 7$

3. Final Answer

Question 1:

i. -0.125

ii. 10

iii. 3

iv. 30

v. 1/5

vi. 5

Question 2:

a.

i. 1

ii. 5/4

iii. 64/125

b. x = 7