Alan

The example in your first paragraph is not consistent with the words! Assuming the example should be 6138 goes to 1386, then:

Prior to the last * the number was 324

The square root of this is 18

Prior to the first * the number was 81.

The original number was therefore the square root of this, namely, 9.

Yes it is different.  It is a computer language convention, from the days when all computer languages were text only, with a limited numbers of characters on a line.  It is often seen as the upper case E as well.

Hmm. If I use LaTeX here in the answer some of it displays properly in the question.  If I delete the LaTeX here the question looks a mess again!!

So:  Use the LaTeX button and copy and paste your questions into the resulting box, having removed the $ signs. e.g.

 If \(z = \frac{ \left\{ \sqrt{3} \right\}^2 - 2 \left\{ \sqrt{2} \right\}^2 }{ \left\{ \sqrt{3} \right\} - 2 \left\{ \sqrt{2} \right\} }\) find \(\lfloor z \rfloor\)

"I did a calculation and got the answer 1.349e-10. Is this the same as 1.349*10^-10? How do I get this value in plain numbers? "

1.349e-10. Is this the same as 1.349*10^-10?  Yes.

Plain number:   Move the decimal point 10 places to the left;  0.000 000 000 134 9

You are correct. I wrote down the wrong number from WolframAlpha!

1) a = 4, b = 9, You should be able to work out c and d easily from this.

3) 722.  I just used WolframAlpha to answer this. (Edited to correct answer - see asdf below).

Like so

Here's a derivation of the relation:

NB For the person who posted the question:  Please note that this is not a proof by induction!

"Solve the quadratic equation by completing the square.

2x^2+2=3x"

Divide through by 2

x^2 + 1 = 3x/2

Rearrange

x^2 - 3x/2 = -1

Add  (1/2*3/2)^2 to both sides:

x^2 - 3x/2 + (1/2*3/2)^2 = (1/2*3/2)^2 - 1  or  x^2 - 3x/2 +9/16 = -7/16

Equation can now be written as (x - 3/4)^2 = -7/16

Hence   x = 3/4 ± √(-7/16)    or  x = (3 ± i√7)/4

.

As follows: