Tetration

$$9-3\frac{1}{3}$$

Transform the mixed numbers into improper fraction. Use this formula (can you see why it's true):

$$a\frac{b}{c}=\frac{ac + b}{c}$$

We get:

$$9 - \frac{3*3+1}{3}=$$

$$9 - \frac{10}{3}=$$

Transform the integer into a fraction with a common denominator (bottom), and calculate the answer:

$$9 - \frac{10}{3}=$$

$$\frac{27}{3}- \frac{10}{3}= \frac{17}{3}$$

Both have to do with round shapes...?

That depends entirely on the calculator.

I have a few old fashion ones  (the ones that are calculators, and nothing more) that can show eight digts, hence the highest # that can be displayed is: 999 999 99.

If scientific notation is "allowed", some calculators can display truly enormous #'s. The calculator on my phone can show numbers up to 10¹⁰⁰⁰. 

The calculator-function on Wolfram Alpha allowes power towers, such as:

$$10^{10^{10^{10^{10}}}}$$

A # so enormous, it is hard to even grasp the concept of. 

Click on the "LaTeX Formul" digital button, then "Operators", then X / Y.    :)

What kind of polyhedron?

An arithmetic sequence has a constant difference between two consecutive terms.

Ex:

1, 5, 9, 13, 17, ...

-46, -44, -42, -40, ...

20, 18, 16, 14, ...

BTW: It is very common to make mistakes based on a plus that should have been a minus or vica versa, especially when you have a minus before a paranthesis. When you have an expression like this, I'll highly recommend solving it this way:

-(x + 3)(2x - 5) = 

-(2x² - 5x +6x -15) =

- 2x² + 5x - 6x + 15 = ...

And NOT this way:

-(x + 3)(2x - 5) = 

- 2x² + 5x - 6x + 15 = ...

Both methods are mathematically correct, but the latter one makes it 1000 times more easy to make a mistake based on a wrong plus/minus. 

General rules:

I: Left, outer, inner, right: (a + b)(c+ d) = ac + ad + bc + bd 

II:  a - b = -(b - a)

III: -(-w) = w

(3x - 2)(4x + 3) - (x + 3)(2x - 5) = (2x + 1)(5x - 2)        Use I:

12x² + 9x - 8x - 6 - (2x² - 5x +6x -15) = 10x²  - 4x + 5x - 2      Use III, add/subtract some x'es and numbers:

12x² + x  - 6 - 2x² + 5x - 6x + 15 = 10x² + x - 2         Add/subtract some x'es and numbers:

.....        ......         .....      .....                                     Add 2 to both sides:

.....        ......    

Can you take over from here?

*** Thx :) ***

Thanks for the answers, and the link :-)

Yes, CPhill, it was something like that I had in mind.

I just don't have a "feel" for modulo, the same way that I have for, say, regular division :)