Rom

$$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{tan}}^{\!\!\mathtt{-1}}{\left(-{\mathtt{1}}\right)} = -{\mathtt{45^{\circ}}}$$

assuming c is real and c > 0

$$\sqrt[6]{c^6}=c$$

Because f(x) is not a 1 to 1 mapping the inverse doesn't exist unless you restrict the domain of x.

if you restrict x to {x:x >= 7} you could find

$$y=x^2+7$$

$$y-7=x^2$$

$$x=\sqrt{y-7}$$

$$f^{-1}(x)=\sqrt{x-7}$$

on the other hand if you restrict x to {x: x <= -7} you get

$$f^{-1}(x)=\sqrt{7-x}$$

$$y=\sqrt{x}+3$$

$$y-3 = \sqrt{x}$$

$$(y-3)^2 = x$$

$$f^{-1}(x)=(x-3)^2$$

in general if you have a point P that you are going to scale about a center of dilation (cx,cy) you end up with

$$P'=sf((P_x-c_x), (P_y-c_y))+(c_x,c_y)=$$

$$(sf(P_x-c_x)+c_x,~sf(P_y-c_y)+c_y)$$

here

$$c_x=c_y=0,~~sf=0.5$$

so we end up with

$$P'=(0.5 P_x, 0.5 P_y)$$

applying that to P=(2,2) we get

$$P'=(0.5\cdot 2, 0.5 \cdot 2)=(1,1)$$

you can figure out the rest of them

$$\sqrt{144 x^2 y^2}=\sqrt{144}\sqrt{x^2}\sqrt{y^2}=12xy$$

you can build this up from the left

one such number is

123654321

$${\frac{{\mathtt{123\,654\,321}}}{{\mathtt{9}}}} = {\mathtt{13\,739\,369}}$$

$${\frac{{\mathtt{12\,365\,432}}}{{\mathtt{8}}}} = {\mathtt{1\,545\,679}}$$

$${\frac{{\mathtt{1\,236\,543}}}{{\mathtt{7}}}} = {\mathtt{176\,649}}$$

$${\frac{{\mathtt{123\,654}}}{{\mathtt{6}}}} = {\mathtt{20\,609}}$$

$${\frac{{\mathtt{12\,365}}}{{\mathtt{5}}}} = {\mathtt{2\,473}}$$

$${\frac{{\mathtt{1\,236}}}{{\mathtt{4}}}} = {\mathtt{309}}$$

$${\frac{{\mathtt{123}}}{{\mathtt{3}}}} = {\mathtt{41}}$$

$${\frac{{\mathtt{12}}}{{\mathtt{2}}}} = {\mathtt{6}}$$

$${\frac{{\mathtt{1}}}{{\mathtt{1}}}} = {\mathtt{1}}$$

if you just work this backwards you can see how it's constructed.  Just make sure your condition is met each time you add a new digit.

based on the observation so far the probabilty of getting a white ball is

$${\frac{{\mathtt{7}}}{{\mathtt{14}}}} = {\mathtt{0.5}}$$

so you'd expect that of then next 18 b***s

$${\mathtt{18}}{\mathtt{\,\times\,}}{\mathtt{0.5}} = {\mathtt{9}}$$

b***s would be white

you do it much like long division of two numbers

       x + 6

       ---------------

x-1 |x^2 + 5x - 1

        x^2 -    x

      ----------------

                   6x - 1

                   6x - 6

                   -------

                           5

so (x^2+5x-1) = (x-1)(x+6)+5

I'm not sure what "conservative" means but 

the 99% confidence margin of error is 

$$1.29\sqrt n$$

and the 95% confidence margin of error is 

$$0.98 \sqrt n$$

with n=450 these numbers are

$${\mathtt{1.29}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{450}}}} = {\mathtt{27.365\: \!032\: \!431\: \!919\: \!389\: \!2}}$$

$${\mathtt{0.98}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{450}}}} = {\mathtt{20.788\: \!939\: \!366\: \!884\: \!497\: \!2}}$$