Math 101 Intermediate Algebra

Order of Operations
Chapter 1, Section 4

Exponents
An exponential expression is bn.

b is called the base and n is called the exponent.

bn means multiply b by itself n times.

A radical expression is .

is the radical sign,
a is the radicand, and
m is the index.

means the square root of a.

The principal (or positive) square root of the number 'a' is written   and is equal to the positive number that when multiplied by itself gives 'a'.

means the cube root of a.

means the mth root of a.

if     .

Order of Operations
These are general steps.

Sometimes the steps must be applied to each term, as in the example in the table below, and then applied to the whole expression.

Example used in table:  evaluate (-5[3 - 6] - 7)2 - (7 - 13) - 34

Steps Example
Remove grouping symbols: (), {}, [].
Work from innermost out.
(-5[-3] - 7)2 - (-6) - 34
(15 - 7)2 + 6 - 34
82 + 6 - 34
Evaluate all terms containing exponents and roots. 64 + 6 - 81
Evaluate all multiplictions and/or divisions in the order they occur, working left to right. Already did this step inside previous terms.
Evaluate all additions and/or subtractions in the order they occur, working left to right. 70 - 81
-11

Note how the steps were followed inside grouping symbols and then followed on the whole expression.

Evaluating Expressions for a Specific Variable Value
This amounts to "plugging it in."

Example: evaluate 10(x - 3)3 - (7 - x2) at x = -5.

 Plug "-5" in for each "x" in the expression 10(-5 - 3)2 - (7 - (-5)2) Evaluate using "order of operations" steps 10(-8)2 - (7 - 25) 10(64) - (-18) 640 + 18 658