Basic Mathematics for College Students

4.6 Square Roots 389

Order of Operations

1. Perform all calculations within parentheses and other grouping symbols
   following the order listed in Steps 2–4 below, working from the innermost
   pair of grouping symbols to the outermost pair.


2. Evaluate all exponential expressions and square roots.


3. Perform all multiplications and divisions as they occur from left to right.


4. Perform all additions and subtractions as they occur from left to right.

EXAMPLE (^4) Evaluate: a. b.
StrategyWe will scan the expression to determine what operations need to be
performed. Then we will perform those operations, one-at-a-time, following the
order of operations rule.
WHYIf we don’t follow the correct order of operations, the expression can have
more than one value.
SolutionSince the expression does not contain any parentheses, we begin with
step 2 of the rules for the order of operations: Evaluate all exponential expressions
and any square roots.
a. Evaluate each square root first.
Do the addition.
b. Evaluate each square root first.
 20 Do the subtraction.

 125  1225  5  15

 11

164  19  8  3

164  19  125  1225

Self Check 4

Evaluate:

a.

b.

Now TryProblems 49 and 53

 19  1196

1121  11

EXAMPLE (^5) Evaluate: a. b.
StrategyWe will scan the expression to determine what operations need to be
performed. Then we will perform those operations, one-at-a-time, following the
order of operations rule.
WHYIf we don’t follow the correct order of operations, the expression can have
more than one value.
SolutionSince the expression does not contain any parentheses, we begin with
step 2 of the rules for the order of operations: Evaluate all exponential expressions
and any square roots.
a.We note that means.
Evaluate the square root first.
Do the multiplication.
b. Evaluate each square root first.
Do the multiplication.
 11 Do the addition.

 20  9

 5116  319 5( 4 )3( 3 )

 60

61100 6( 10 )

61100 6  1100

61100  5116  319

Self Check 5

Evaluate:

a.

b.

Now TryProblems 57 and 61

 6125  2136

81121

EXAMPLE (^6) 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 rule, we must work from the innermostpair of
grouping symbols to the outermost.

12  3 C 32 (41) 136 D

Self Check 6

Evaluate:

Now TryProblems 65 and 69

10  4 C 22 (32) 14 D