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Parentheses, Brackets, Braces etc. Parentheses ( ) are used to group numbers or variables. Everything inside parentheses must be done before any other operations. Note that any expression within parentheses ( ) should be regarded as equivalent to a single number. Thus 5 ´ ( 7 + 9 ) means the sum ( 7 + 9 = 16 ) is to be multiplied by 5 ; hence 5 ´ ( 7 + 9 ) = 5 ´ ( 16 ) = 80. If there is no bracket, we would have 5 ´ 7 + 9 = 35 + 9 = 44. Again, ( 8 + 3 ) ´ ( 9 - 5 ) = (11) ´ ( 4 ) = 44. Just as 9 ´ a is written as 9a , similarly 5 ´ ( a + b ) is written as 5 ( a + b ). When a parentheses is preceded by a - sign, to remove parentheses, change the sign of each term within it. For example 8 - ( a + 3 b + 4 - c ) = 8 - a - 3 b - 4 + c = 8 - 4 - a - 3 b + c = 4 - a - 3 b + c
Sometimes a line segment is drawn over the expression which is to be treated as a single number. It serves the same purpose as parenthesis. Thus a + b ´ c + d is the same as ( a + b ) ´ ( c + d ). This line segment so used is called a Vinculum. Brackets [ ] and braces { } are also used to group of numbers or variables. Technically, they are used after parentheses. The order is parentheses - brackets - braces. In short { [ ( ) ] }. Sometimes, instead of brackets or braces, we can use larger parentheses. For example, ( 4 . ( 3 + 7 ) ) + 8 Using all three grouping symbols, we will have a number, looking like { 3 - [ 5 ( 3 + 7 ) - 1 ] }.
Solution: = 7 { 9 + [ 4 ( 6 + 3 ) - 5 ] } = 7 { 9 + [ 4 ( 9 ) - 5 ] }......( 6 + 3 ) = 9 = 7 { 9 + [ 36 - 5 ] } = 7 { 9 + 31 }......[ 36 - 5 ] = 31 = 7 ´ 40......{ 9 + 31 } = 40 = 280
Exponents and Powers Positive integers are used as exponents to simplify repeated products such as x2 = x . x , x3 = x . x . x , x4 = x . x . x . x ....... In general, if n is a positive integer, xn = x . x . x .......x
Here the number ‘ x ’ is called the ‘ base ’ and the positive integer ‘ n ’ is called the ‘exponent’. Read xn as " x to the nth power " OR " x is raised to power n " OR " x to the nth
Note that x1 = x. Some examples are
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