Question

How do you simplify `n-{1-[n-(1-n)-1]}`?

Answer 1

`n-{1-[n-(1-n)-1]} = 3n-3`

Explanation:

Given:

`n-{1-[n-(1-n)-1]}`

Work from the inside, outwards, replacing each parenthesised expression by its simplest form and evaluating left unparenthesised compound expressions from left to right...

`n-{1-[n-(1-n)-1]} = n-{1-[n+(n-1)-1]}`

`color(white)(n-{1-[n-(1-n)-1]}) = n-{1-[n+n-1-1]}`

`color(white)(n-{1-[n-(1-n)-1]}) = n-{1-[2n-1-1]}`

`color(white)(n-{1-[n-(1-n)-1]}) = n-{1-[2n-2]}`

`color(white)(n-{1-[n-(1-n)-1]}) = n-{1+[2-2n]}`

`color(white)(n-{1-[n-(1-n)-1]}) = n-{1+2-2n}`

`color(white)(n-{1-[n-(1-n)-1]}) = n-{3-2n}`

`color(white)(n-{1-[n-(1-n)-1]}) = n+{2n-3}`

`color(white)(n-{1-[n-(1-n)-1]}) = n+2n-3`

`color(white)(n-{1-[n-(1-n)-1]}) = 3n-3`

Alternatively, if you wanted to find the result as quickly as possible, you could note that the whole given expression is a sort of sum, with the minus signs toggling the value of the individual terms between `+` and `-`. So you could look at all the terms involving `n`, noting whether they will end up as positive or negative, and add them together. Then do the same with the constant terms...

The first `n` is positive:

`color(red)(n)-{1-[n-(1-n)-1]}`

The second `n` is also positive, since it is affected by an even number (`2`) of minus signs:

`ncolor(red)(-){1color(red)(-)[color(red)(n)-(1-n)-1]}`

The third `n` is also positive, since it is affected by an even number (`4`) of minus signs:

`ncolor(red)(-){1color(red)(-)[ncolor(red)(-)(1color(red)(-n))-1]}`

So the final term in `n` is `3n`.

Similarly we can find that all the `1`'s are affected by an odd number of minus signs, resulting in a final constant term `-3`.

Answer 2

`3n-3`

Explanation:

Start with the innermost brackets and work outwards, one set of brackets at a time.
Remember that a negative in front of a bracket changes the sign when you multiply into the bracket.

`n-{1-[ncolor(blue)(-(1-n))-1]}" "larr` remove the round brackets

`=n-{1-[ncolor(blue)(-1+n)-1]}`

`=n-{1color(red)(-[n-1+n-1]}" "larr` remove square brackets

`=n-{1color(red)(-n+1-n+1]}`

`=ncolor(green)(-{1-n+1-n+1})" "larr` remove curly brackets

`=n-1+n-1+n-1" "larr` simplify

`=3n-3`