Dr. Algebra

When you have a generic expression or equation, its often helpful to be able to evaluate or simplify it.  Knowing what order things like parentheses, brackets, and exponents, should be evaluated will help you to obtain a correct result. 

Example #1 — Evaluate the expression [katex]3( x+4)[/katex]  when x is 5.

(Correct Solution):  [katex]3( x+4)\to 3\left( 5+4 \right)\to3\left( 9\right)\to27[/katex]

(Incorrect Solution):  [katex]3\left( x+4 \right) \to 3\left( 5+4 \right) \to 15+4 \to 19[/katex]

Can you see the difference in the correct and incorrect solution?  Knowing how to handle parenthesis along with multiplication and addition is important and can give incorrect results if not performed correctly. 

To provide consistency to evaluating expressions, a standard “Order Of Operations” has been developed.  When the standard is correctly and consistently applied, it will always result in an unambiguous and correct answer.

Evaluating expressions MUST be performed in the following order (and note it is possible to nest operations).

• 1. Anything in ( ), or [ ], must be fully evaluated prior to anything outside the ( )
  2. Exponents are first.
  3. Multiplications and divisions are performed next
     
     • Use LEFT to RIGHT if there are more than two terms.
  4. Additions and subtractions are ALWAYS last.

Using the division symbol ‘÷’ is very likely to cause confusion, especially if parenthesis are not used.  A better way is to use a horizontal fraction bar.

Example: Instead of writing [katex]6÷2\cdot 3[/katex] you should use [katex]\frac{6}{2}\cdot 3[/katex] or [katex]\frac{6}{2\cdot 3}[/katex] — Whichever you intend to convey.  Can you see the difference?