Powers, Exponents, and Roots

Operations with Exponents

Order of Operations with Exponents

In the SparkNote on Operations, we learned the order of operations, which uses these steps:

Step 1. Carry out the operations within the parentheses.
Step 2. Multiply and divide (it does not matter which comes first).
Step 3. Add and subtract (it does not matter which comes first).

But where do calculations with exponents fit into the picture?

The new order of operations, including operations with exponents, is:

Step 1. Carry out the operations within the parentheses (or absolute value).
Step 2. Evaluate powers and roots.
Step 3. Multiply and divide (it does not matter which comes first)--this includes applying a minus sign.
Step 4. Add and subtract (it does not matter which comes first).

Note that taking a power always comes before applying a minus sign unless the sign is inside the parentheses which form the base number. Observe:

-7² = - 49
(- 7)² = 49
-2⁴ = - 16
(- 2)⁴ = 16
-5³ = - 125
(- 5)³ = - 125 (because 3 is odd)
- = - 4
is undefined (we cannot take the square root of a negative number)