If an expression contains more than one pair of grouping symbols, we always
begin by working within the innermost pairand then work to the outermost pair.
Innermost parenthesesOutermost brackets 16 6 [ 42 3 ( 5 2 )]
106 Chapter 1 Whole Numbers
The Language of Mathematics Multiplication is indicated when a number is
next to a parenthesis or a bracket. For example,Multiplication Multiplication 16 6[4^2 3(52)]
EXAMPLE (^8) Evaluate:
StrategyWe will work within the parentheses first and then within the brackets.
Within each set of grouping symbols, we will follow the order of operations rule.
WHYBy the order of operations, we must work from the innermostpair of
grouping symbols to the outermost.
Solution
Do the subtraction within the
parentheses.
Evaluate the exponential
expression:
Do the multiplication within the
brackets.
Do the subtraction within the
brackets.
Do the multiplication:
58 Do the addition.
16 42 6[7]42.
16 6[7]
16 6[169]
42 16.16 6[163(3)]
16 6[4^2 3( 5 2 )] 16 6[4^2 3( 3 )]
16 6[4^2 3(52)]
Self Check 8
Evaluate:Now TryProblem 43130 7[2^2 3(62)]
Caution! In Example 8, a common mistake is to incorrectlyadd 16 and 6
instead of correctlymultiplying 6 and 7 first. This error produces a wrong
answer, 154.154
22[7]
16 6[7]
16 6[169]
16 6[163(3)]
16 6[4^2 3(52)] 16 6[4^2 3(3)]
EXAMPLE 9
Evaluate:StrategyWe will evaluate the expression above and the expression below the
fraction bar separately. Then we will do the indicated division, if possible.
WHYFraction bars are grouping symbols. They group the numerator and
denominator. The expression could be written [2(13)2)][3(2^3 )].2(13) 2
3(2^3 )
Self Check 9Evaluate:Now TryProblem 47