The problem consists of 8 sub-problems. Each sub-problem contains an equation and a variable in parentheses. The goal is to solve for the variable in parentheses in terms of the other variables.

1. Problem Description

The problem consists of 8 sub-problems. Each sub-problem contains an equation and a variable in parentheses. The goal is to solve for the variable in parentheses in terms of the other variables.

2. Solution Steps

1. $A = \pi r^2$ for $r$.

Divide both sides by $\pi$:

$\frac{A}{\pi} = r^2$

Take the square root of both sides:

$r = \sqrt{\frac{A}{\pi}}$

2. $v = u + at$ for $t$.

Subtract $u$ from both sides:

$v - u = at$

Divide both sides by $a$:

$t = \frac{v-u}{a}$

3. $v^2 = u^2 + 2as$ for $a$.

Subtract $u^2$ from both sides:

$v^2 - u^2 = 2as$

Divide both sides by $2s$:

$a = \frac{v^2 - u^2}{2s}$

4. $F = \frac{Gm}{r^2}$ for $m$.

Multiply both sides by $r^2$:

$Fr^2 = Gm$

Divide both sides by $G$:

$m = \frac{Fr^2}{G}$

5. $V = \frac{4}{3} \pi r^3$ for $r$.

Multiply both sides by $\frac{3}{4}$:

$\frac{3}{4}V = \pi r^3$

Divide both sides by $\pi$:

$\frac{3V}{4\pi} = r^3$

Take the cube root of both sides:

$r = \sqrt[3]{\frac{3V}{4\pi}}$

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

Multiply both sides by $6x+3$:

$b(6x+3) = 2x - 2$

$6bx + 3b = 2x - 2$

$6bx - 2x = -2 - 3b$

$x(6b - 2) = -2 - 3b$

Divide both sides by $(6b - 2)$:

$x = \frac{-2 - 3b}{6b - 2} = \frac{-(3b+2)}{2(3b-1)} = -\frac{3b+2}{2(3b-1)}$

7. $m^2 + 4qp = n + x$ for $q$.

Subtract $m^2$ from both sides:

$4qp = n + x - m^2$

Divide both sides by $4p$:

$q = \frac{n + x - m^2}{4p}$

8. $T = \sqrt{\frac{e}{g}}$ for $g$.

Square both sides:

$T^2 = \frac{e}{g}$

Multiply both sides by $g$:

$gT^2 = e$

Divide both sides by $T^2$:

$g = \frac{e}{T^2}$

3. Final Answer

1. $r = \sqrt{\frac{A}{\pi}}$

2. $t = \frac{v-u}{a}$

3. $a = \frac{v^2 - u^2}{2s}$

4. $m = \frac{Fr^2}{G}$

5. $r = \sqrt[3]{\frac{3V}{4\pi}}$

6. $x = -\frac{3b+2}{2(3b-1)}$

7. $q = \frac{n + x - m^2}{4p}$

8. $g = \frac{e}{T^2}$