Simplify the following expression: #101-{[(110 -: 2)-: 11]xx(10+4xx2) +7}+ [8xx(20 -: 5-1)-3xx3] -: 5#?

Wow... that's one big equation. Let's take this step by step.

First we'll start with the original:

#101-{[(110-:2)-:11]xx(10+4xx2)+7}+[8xx(20-:5-1)-3xx3]-:5#

Before we dive into this thing, let's look at the structure - there is #101# - big brackets + smaller brackets#-:5#. PEDMAS has us work things in brackets (Parentheses) first and since the big brackets and the smaller brackets are separated by the #+#, we can work them separately. I'm going to simplify the big brackets first:

#{[(110-:2)-:11]xx(10+4xx2)+7}#

There are brackets (and brackets within brackets) in this, so I'm going to work those first. There are 2 sets here and I'll work them side by side. In this first step, let's do the division and in the second set we have both addition and multiplication - so we'll do the multiplication first:

#{[55-:11]xx(10+8)+7}#

I can now do the next division in the first bracket and do the addition in the second:

#{5xx18+7}#

We'll finish this up with the multiplication first, then the addition:

#90+7=97# which I'll substitute back into our original:

#101-97+[8xx(20-:5-1)-3xx3]-:5#

Let's now work that second bracket:

#[8xx(20-:5-1)-3xx3]#

There's a bracket in here that we need to work first. Within that bracket there is both division and subtraction - we'll divide first:

#[8xx(4-1)-3xx3]#

and now the subtraction:

#[8xx3-3xx3]#

We now have 2 multiplications and a subtraction, so we'll do the multiplications first:

#[24-9]=15#

Let's substitute that into the original:

#101-97+15-:5#

Almost there! We have subtraction, addition, and division. We'll do the division first:

#101-97+3#

and now the subtraction and addition:

#4+3=7#

Count the number of terms and work through each one carefully.
Each term must give a number answer.

There are only 3 terms in this expression:

#color(magenta)(101)color(blue)(-{[(110-:2)-:11]xx(10+4xx2)+7})color(green)(+[8xx(20-:5-1)-3xx3]-:5)#

Let's handle one at a time.
The first is easy. it is #color(magenta)(101)#.

#color(blue)(-{[color(red)((110-:2))-:11]xx(10+color(red)(4xx2))+7})#

#"parentheses first, but remember to do the multiplication "#
#"and division before addition and subtraction"#

#color(blue)(-{[color(red)(55)-:11]xx(10+color(red)(8))+7})#

#color(blue)(-{[color(red)(5]xx(color(red)(18))+7})#

#color(blue)(-[color(red)(90+7)] = -97#

Now for the third term:

#color(green)(+[8xx(20-:5-1)-3xx3]-:5)#

#color(green)(+[8xxcolor(red)((20-:5-1))-color(orange)(3xx3)]-:5)#

#color(green)(+[8xxcolor(red)((4-1))-color(orange)(9)]-:5)#

#color(green)(+[8xxcolor(red)(3)-color(orange)(9)]-:5)#

#+[color(red)(24)-color(orange)(9)]-:5)#

#+color(green)(15-:5) = color(green)3#

the whole expression simplifies to
#color(magenta)(101)color(blue)(-97)color(green)(+3)#

=#color(magenta)(101)color(green)(+3)color(blue)(-97)#

=#7#