In this publication, we will consider the basic rules for opening brackets, accompanying them with examples for a better understanding of the theoretical material.
Bracket expansion – replacement of an expression containing brackets with an expression equal to it, but without brackets.
Bracket expansion rules
Rule 1
If there is a “plus” before the brackets, then the signs of all numbers inside the brackets remain unchanged.
Explanation: Those. Plus times plus makes a plus, and plus times a minus makes a minus.
examples:
6 + (21 – 18 – 37) =6 + 21 – 18 – 37 20 + (-8 + 42 – 86 – 97) =20 – 8 + 42 – 86 – 97
Rule 2
If there is a minus in front of the brackets, then the signs of all numbers inside the brackets are reversed.
Explanation: Those. A minus times a plus is a minus, and a minus times a minus is a plus.
examples:
65 – (-20 + 16 – 3) =65 + 20 – 16 + 3 116 – (49 + 37 – 18 – 21) =116 – 49 – 37 + 18 + 21
Rule 3
If there is a “multiplication” sign before or after the brackets, it all depends on what actions are performed inside them:
Addition and/or subtraction
a ⋅ (b – c + d) =a ⋅ b – a ⋅ c + a ⋅ d (b + c – d) ⋅ a =a ⋅ b + a ⋅ c – a ⋅ d
Multiplication
a ⋅ (b ⋅ c ⋅ d) =a ⋅ b ⋅ c ⋅ d (b ⋅ c ⋅ d) ⋅ a =b ⋅ с ⋅ d ⋅ a
Division
a ⋅ (b : c) =(a ⋅ b) : p =(a : c) ⋅ b (a : b) ⋅ c =(a ⋅ c) : b =(c : b) ⋅ a
examples:
18 ⋅ (11 + 5 – 3) =18 ⋅ 11 + 18 ⋅ 5 – 18 ⋅ 3 4 ⋅ (9 ⋅ 13 ⋅ 27) =4 ⋅ 9 ⋅ 13 ⋅ 27 100 ⋅ (36 : 12) =(100 ⋅ 36) : 12
Rule 4
If there is a division sign before or after the brackets, then, as in the rule above, it all depends on what actions are performed inside them:
Addition and/or subtraction
First, the action in parentheses is performed, i.e. the result of the sum or difference of numbers is found, then division is performed.
a : (b – c + d)
b – с + d = e
a : e = f
(b + c – d) : a
b + с – d = e
e : a = f
Multiplication
a : (b ⋅ c) =a : b : c =a : c : b (b ⋅ c) : a =(b : a) ⋅ p =(with : a) ⋅ b
Division
a : (b : c) =(a : b) ⋅ p =(c : b) ⋅ a (b : c) : a =b : c : a =b : (a ⋅ c)
examples:
72 : (9 – 8) =72:1 160 : (40 ⋅ 4) =160: 40: 4 600 : (300 : 2) =(600 : 300) ⋅ 2