Evaluate the following expression: #4-7-:(3+1)-3-2*(5-9)+1# By how much does the value of the expression change if the parentheses are removed?

The best way to approach calculations involving mixed operations, is to realise that there are "powerful" operations and "weak" operations.
The strongest operations are powers and roots ,
then come multiplication and division ;
the weakest are addition and subtraction.
They are therefore done in this order.
However, sometimes a weaker operation must be done first and that is indicated with parentheses, or brackets.

ALWAYS count the number of terms first!! (They are separated by the + and - signs).
There must be a final answer for each term, only then can they be added or subtracted - usually working from left to right, although this can be changed, using the commutative law, to make computation easier.

This is what is indicated by BODMAS, PEDMAS, etc.

In # color(green)(4) color(blue)(- 7 ÷ (3 + 1)) color(red)( - 3) color(turquoise)( - 2 xx (5 - 9)) color(orange)(+ 1)# there are 5 terms
#rArr color(green)(4) color(blue)(- 7/4) color(red)( - 3) color(turquoise) ( - 2xx(-4)) color(orange)( + 1) #

#rArrcolor(green)(4) color(turquoise) ( +8) color(orange)( + 1) color(red)( -3) color(blue)(- 1 3/4) #

#= 10 color(blue)(-1 3/4)#

= # 8 1/4#

Without the parentheses, there are 7 terms:

# color(green)(4) color(blue)( - 7 ÷ 3) + 1 color(red)( - 3) color(turquoise)( - 2 xx 5) - 9 color(orange)( + 1)#
= # 4 - 7/3 + 1 - 3 - 10 - 9 + 1#
= # 4 + 1 +1 - 3 - 10 - 9 - 2 1/3#
= #6 - 24 1/3#

= #-18 1/3#

The difference between the two answers is
#8 1/4 - (-18 1/3)" # # rArr# #8 1/4 + 18 1/3#

= #26 (3 + 4)/12#

= #26 7/12#