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## SYMBOLS OF GROUPINGS

Symbols of groupings are used to multiply. This is to separate item-to-item. The first to manipulate is the innermost part of the expression, say 2[-3(4)], we mean -3(4) as the innermost term as the first one to operate. So we can say that 2[-3(4)] is now 2[-12]. We remove the symbol of grouping as we do the operation of multiplication simultaneously.Then one by one from the innermost part of the expression we actually moving outward direction to eliminate the symbols of groupings.

Illustration:

SYMBOL                              NAME  Vinculum

(              )                          Parentheses

[              ]                          Brackets

{              }                          Braces

Example:

1. 2 (3)    means 2 multiplied by 3 so it gives 6.

2.       -3 ( 4 ) means -3 multiplied by 4 so it gives – 12.

3.       5 [ 2 ] means 5 multiplied by 2 so it gives 10.

4.       4 { - 5 } means 4 multiplied by 5 so it gives 20.

We use several symbols of groupings to separate terms. From vinculum, or parentheses as majority likes it,up to braces.

Learn from this illustration. { 2 [ 5 -2( 3 (4) ) -1] +6}  To treat this problem, perform the operation from the innermost part of the expression. So
the  3(4)is the innermost part and is equal to 12, and it should be done first to have 12 for the innermost result. Thus the next look of the expression will resulting to { 2 [ 5 -2( 12 ) -1] +6} as initial result. The second innermost part of { 2 [ 5 -2( 12 ) -1] +6} is the -2(12) and it gives us – 24. Now by following the first mathematical manipulation, the new set up will became { 2 [ 5 -24 -1] +6}. The 3rd innermost part is the [ 5 -24 -1] which gives us – 20. Now, finally, the whole set up will become { 2 [-20] +6}, so gives us final answer when performed, {-40 + 6} = -34. In short, the final answer to the given { 2 [ 5 -2( 3 ( 4)  ) -1] +6} is -34. If we will do this in algebraic manner, the gradual development is illustrated.

1. { 2 [ 5 -2( 3 (4)  ) -1] +6} = { 2 [ 5 -2( 12 ) -1] +6}

= { 2 [ 5 -24 -1] +6}

= { 2[ -20] +6}

= - 40 + 6

2. {-3+[2-(2-3+4)-5]-1} + [-(2-5)-4] = (the innermost part was highlighted)

= {-3+[2-(2-3+4)-5]-1+ [-(2-5)-4]

{-3+[2-(6)-5]-1} + [-(-3)-4] =

{-3+[2-(6)-5]-1} + [-(-3)-4] =

= {-3+[2-6-5]-1} + [+3-4] = (parenthesis was eliminated)

{-3+[2-6-5]-1} + [+3-4] =

{-3+[-9]-1} + [-1] =

{-3-9-1} -1 = (brackets was eliminated)

{-3-9-1-1 = (-3-9-1 is now the last innermost part)

= -13 -1(braces was eliminated)

= -14, (finally, we arrived to the final answer)

Try this simple exercises.

Direction:  Remove the symbol of groupings & simplify.

1. -3{ -2 [ 3 – (2 – 1) -2 ] + 1 }

2.  9 { 2/3 [ ( 2 – 3 ) -2 ] }

3.  a b ( a2) + a ( a2 b )

4.  (2 x3 y4 z3 ) / ( 4 x-1 y-2 z )

5.  1 – 2 – 3 – 4 – 5 – 6 – 7 – 8 – 9 ( 0 )

More exercises.

Simplify the given expression.

1.      22 – 33 + 4( 32 – 23) – 1

2.      16-20.  ( 2/3)2 / (3/2)2 + 2( - 1)

3.       ( -15 – 32)) – (1 – 2 – 4 + 42)

4.      5 – 7 + 3 (8 - 2) + 2 – ½

5.       22/4 ( 3 / 6 + 1) – 2 – 2

6.      -3{ -2 [ 3 – (2 – 1) -2 ] + 1 }

7.       (2 x3 y4 z3 ) / ( 4 x-1 y-2 z )

8.       9 { 2/3 [ ( 2 – 3 ) -2 ] }

9.       8.   1 – 2 – 3 – 4 – 5 – 6 – 7 – 8 – 9 ( 0 )

10.    a b ( a2) + a ( a2 b )

11. 3(5 − x) − 2(5 + x) + 3(x + 1)

12.  5(2 − x) − 3(4 − 2x) - 20

13.  3m + 12 ( 2(m − 3) + 4 )

14.  x / 5 + x / 3 + 10

15.  (x+1) / 4 - 5

1. { -3 [ - 4 +2 ( 3 -  4  ) -5 ]  + 6 } + 12 =

2.  { - [ 3 (2 - 5)-3 ] +2 } -  { -3 [ - 4 + 2 ( 3 -  4  ) -5 ]  + 6 } + 12 =

3.  - { -5 - [ -4 (-2) -6 ] + 4 } - { - [ -4 (-6 + 6)+8 ] -1 } +  { -9 [  9 + 2 ( -3 + 9  ) +9 ]  - 6 } - 9 =

4.  { -3 / 2 [ - 4 / 4 +2 ( - 3 / 2 -  4 ) - 5 ]  + 6 } + 1 =

More problems on symbols of groupings.

Perform the indicated operations.

1.       – { - a [ -2 ( -a )-5 ]-7 }=

2.       – { - ½ [ ¼ (-2)-9] -3} – { -2[ - ¾ ( -1/2) + 2/3 } =

3.       - x2[ - x3(2x)-3]=

4.       –4y3-{2x4 –[ 3x4+(7x4) –y2(y)] -5} =

5.       –3y5-x{7x1/2 –x2[ 2x2/3+x1/2(6x3/4) –5y2(y2/5)] -5}=

6.        - {2x4y3 –x2[ 3x4y1/2+x6/5(5x5/6) –y2/5(y7/8)] -7}=

7.       – 5a-{2ax4y3 –x2[ 3ax4y1/2+(5ax5/6) –ay2/5(ay7/8)] -9}=

8.       - {5y(3y3) –x2/5[ 3x4y1/3+(5x5/6) –y3/5(y8/8)] +8}=

9.       - {12x1/3-[-(-2x)x3-4]-y3} –x2-x{[ -5x3/4y1/2-(8x5/6) +4y3/5(-3y7/8)] -4}=

10.        –5b- 2{b2/5+[-3c2(-8b)-2c3]c3} +3b2-4c{[ -5c3/4b1/2-(8b5/6) +5c3/5(-3c7/8)] -1}

More on reference below: