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# Grouping Symbols

There are three main types of grouping symbols:

 ( ) Parantheses [ ] Square brackets { } Braces

When an expression involves more than one grouping symbol, we start from the inner most grouping symbol first. Let’s look at some examples –

1.  4 x [5 x (60 – 10)]

=  4 x [5 x 50]

=  4 x 250

=  1000

2. {2 x [5 + 6(2 + 3)]}  ÷  (5 + 2)

=  {2 x [5 + 6(5)]}  ÷  (7)

=  {2 x [5 + 30]}  ÷  7

=  {2 x 35}  ÷  7

=  70  ÷  7

=  10

There are three main laws in grouping symbols:

### Cumulative law

This is used mainly for addition and multiplication, and states that the order of the numbers can be interchanged without changing the answer.

For example 2 + 5 + 8 =   5 + 2 + 8 =  5 + 10  =  15

2 x 3 x 5  =  2 x 3 x 5  =  2 x 15  =  30

### Associative law

This is again used for addition and multiplication. It states that the numbers can be grouped in a different way without changing the answer.

For example (5 + 3) + 7 =   5 + (3 + 7) =  5 + 10  =  15

(5 x 3) x 6  =  5 x (3 x 6)  =  5 x 18  =  90

### Distributive law

This law is used to expand an expression by removing the grouping symbol, by multiplying the number outside the grouping symbol by each term inside.

For example 5 x (210 – 122)  =  5 x 210 – 5 x 122

=  1050 – 610  =  440

27 x 102  =  27(100 + 2)

=  2700 + 54  =  2754