Order of operations

Many students learn the order of operations in math as an acronym (PEMDAS) or as the phrase Please Excuse My Dear Aunt Sally. This stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.  It would actually be better written as

P: Parentheses
E: Exponents
MD: Multiplication and Division
AS: Addition and Subtraction

In fact, multiplication does NOT come before division. Likewise, addition does NOT come before subtraction.  If you have done all the parentheses and exponents and there are both multiplications and divisions to be performed, they should be done from left to right. Likewise, if everything else has been completed and it's time to do additions and subtractions, perform the operations from left to right.

Here are some numbers in a string of operations. I will do the operations step by step with the next operation to perform in bold, and an explanation of why.

sqrt((6-2)² + (10-4)²)
sqrt((6-2)² + (10-4)²) Inside parentheses first, so 6-2 becomes 4.
sqrt(4² + (10-4)²) The second set of inside parentheses first, so 10-4 becomes 6.

sqrt(4² + 6²) We need to simplify what is inside the parentheses. Exponents before addition, so 4² = 16
sqrt(16 + 6²) Now the other exponent, 6² = 26
sqrt(16 + 36) Now addition is the last thing inside the parentheses, 16 + 36 = 52
sqrt(52) ~= 7.211.  52 = 2*2*13, so the simplified square root is 2*sqrt(13)