Rules for expanding brackets with examples

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.

Content

Bracket expansion rules

Rule 1

If there is a “plus” before the brackets, then the signs of all numbers inside the brackets remain unchanged.

a + (b – c – d + e) = a + b – c – d + e

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.

a – (b – c – d + e) = a – b + c + d – e

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

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