Question 117895
<pre><font face = "book antiqua" size = 4><b>
2{[3(m-5)+18]-[3(5m-3)+4]}

First look for the first innermost pair of grouping symbols.
That is, grouping sysmbols which have no grouping symbols 
between them.  So we see (m-5).

Then we ask: "Is there anything which we can do with what is
inside " m-5 ". 

The answer is no.  

Then we ask: "How do we remove those grouping symbols around
" m-5 "?"

The answer is by distributing the 3 just outside the (m-5),
getting 3m-15, so in place of 3(m-5) we write 3m-15, and
copy everything else over, getting

2{[3m-15+18]-[3(5m-3)+4]}

-----------------------------------------

Now we start over. Look for the first innermost pair of 
grouping symbols. That is, grouping sysmbols which have 
no grouping symbols between them.  So we see [3m-15+18].

Then we ask: "Is there anything which we can do with what is
inside "3m-15+18". 

The answer is yes, we can combine the -15 and the +18 getting
"3m+3".  So we replace "3m-15+18" by "3m+3", and copy 
everything else over:

2{[3m+3]-[3(5m-3)+4]}   

---------------------------------------

Now we start over. Look for the first innermost pair of 
grouping symbols. That is, grouping sysmbols which have 
no grouping symbols between them.  So we see [3m+3].

Then we ask: "How do we remove those grouping symbols around
3m+3?"

The answer is by putting the invisible 1 just left of [3m+3],
like this:

2{1[3m+3]-[3(5m-3)+4]}

and distributing the 1 just outside the [3m+3],
getting 3m+3, so in place of [3m+3] we write 3m+3, and
copy everything else over, getting

2{3m+3-[3(5m-3)+4]}

---------------------------------------

Now we start over again. Look for the first innermost pair of 
grouping symbols. That is, grouping symbols which have 
no grouping symbols between them.  So we see (5m-3).

Then we ask: "Is there anything which we can do with what is
inside, that is, "5m-3". 

The answer is no.  

Then we ask: "How do we remove those grouping symbols around
" 5m-3 "?

The answer is by distributing the 3 just outside the (5m-3),
getting 15m-9, so in place of 3(5m-3) we write 15m-9, and
copy everything else over, getting

2{3m+3-[15m-9+4]}

--------------------------------------------

Once more, we start over. Look for the first innermost pair 
of grouping symbols. That is, grouping sysmbols which have 
no grouping symbols between them.  So we see [15m-9+4].

Then we ask: "Is there anything which we can do with what is
inside, namely, "15m-9+4". 

The answer is yes. The answer is yes, we can combine the -9 
and the +4 getting "15m-5".  So we replace "15m-9+4" by "15m-5",
and copy everything else over:

2{3m+3-[15m-5]}   

-----------------------------

Once more, we start over. Look for the first innermost pair 
of grouping symbols. That is, grouping sysmbols which have 
no grouping symbols between them.  So we see [15m-5].

Then we ask: "Is there anything which we can do with what is
inside, namely, "15m-5". 

The answer is no.

Then we ask: "How do we remove those grouping symbols around
15m-5?"

The answer is by putting the invisible 1 just left of [15m-5],
like this:

2{3m+3-1[15m-5]}

and distributing the -1 just outside the [15m-5],
getting -15m+5, so in place of -1[15m-5] we write -15m+5, and
copy everything else over, getting

2{3m+3-15m+5}

--------

Now we start over again. Look for the first innermost pair of 
grouping symbols. That is, grouping symbols which have 
no grouping symbols between them.  So we see {3m+3-15m+5}.

Then we ask: "Is there anything which we can do with what is
inside, namely " 3m+3-15m+5 ". 

The answer is yes, we can combine the 3m and the -15m, and also
the +3 and the +5, getting "-12m+8".  So we replace "3m+3-15m+5"
by "-12m+8", and copy everything else over:

2{-12m+8}   

---------------------------------------

Starting over again, we look for the first innermost pair of 
grouping symbols. That is, grouping sysmbols which have no 
grouping symbols between them.  So we see {-12m+8}.

Then we ask: "Is there anything which we can do with what is
inside, namely, " -12m+8 ". 

The answer is no.  

Then we ask: "How do we remove those grouping symbols around
" -12m+8 "?"

The answer is by distributing the 2 just outside the {-12m+8},
getting -24m+16, so in place of 2{-12m+8} we write -24m+16.

There are no grouping symbols and nothing else can be done with
-24m+16, so we are done.

The answer is -24m+16.

Edwin</pre>