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# Precedence Help

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By McGraw-Hill Professional
Updated on Sep 15, 2011

## Precedence

Mathematicians, scientists, and engineers have all agreed on a certain order in which operations should be performed when they appear together in an expression. This prevents confusion and ambiguity. When various operations such as addition, subtraction, multiplication, division, and exponentiation appear in an expression and you need to simplify that expression, perform the operations in the following sequence:

• Simplify all expressions within parentheses from the inside out.
• Perform all exponential operations, proceeding from left to right.
• Perform all sums and differences, proceeding from left to right.

Here are two examples of expressions simplified according to these rules of precedence. Note that the order of the numerals and operations is the same in each case, but the groupings differ.

[(2 + 3) (−3 − 1) 2 ] 2

[5 × (−4) 2 ] 2

(5 × 16) 2

80 2

6400

{[2 + 3 × (−3) − 1] 2 } 2

{[2 + (−9)−1] 2 } 2

(−8 2 ) 2

64 2

4096

Suppose that you’re given a complicated expression and that there are no parentheses, brackets, or braces in it? This does not have to be ambiguous as long as the preceding rules are followed. Consider this example:

z = −3 x 3 + 4 x 2 y − 12 xy 2 − 5 y 3

If this were written with parentheses, brackets, and braces to emphasize the rules of precedence, it would look like this:

z = [−3( x 3 )] + {4 [( x 2 ) y ]} − {12 [ x ( y 2 )]} − [5 ( y 3 )]

Because we have agreed on the rules of precedence, we can do without the parentheses, brackets, and braces. This will help to keep pure mathematicians happy. It might someday keep one of your research projects from going awry! Nevertheless, if there is any doubt about a crucial equation, you’re better off to use a couple of unnecessary parentheses than to make a costly mistake.

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