Algebra order of operations

July 18, 2014, Grade 5-6 Algebra Mixed Review 4.
The following example shows how to simplify an algebraic expression step by step using correct order of operations in algebra.
Order of operations:
1. Parentheses. Simplify all operations within a pair of parentheses until the result is a number (integer, whole number, decimal, fraction) or cannot be simplified further. If there are nested parentheses, start with the innermost pair of parentheses.
2. Power or exponent.
3. Multiply or Divide.
4. Add or Subtract.
5. Perform operations left to right to combine two terms into one, once the operations are at the same level.
\(2^3+3^2-5\left(4+\dfrac{2}{3}\right)-\left(\dfrac{3}{4}\right)^3+12\left(1-\dfrac{2}{3}\right)\)
Step 1: Parentheses. Look for each pair of parentheses that can simplify its expression further or still need to perform algebraic operations. The expression \(\left(\dfrac{3}{4}\right)\) is already complete at this stage since inside this pair of parentheses is a fraction. We only need to perform an addition inside \(\left(4+\dfrac{2}{3}\right)\) and a subtraction inside \(\left(4+\dfrac{2}{3}\right)\) as following:
\(2^3+3^2-5\boxed {\left(4+\dfrac{2}{3}\right)}-\left(\dfrac{3}{4}\right)^3+12 \boxed{\left(1-\dfrac{2}{3}\right)}\)
Operations required in step 1:
\(4+\dfrac{2}{3}=\dfrac{12}{3}+\dfrac{2}{3}=\dfrac{14}{3}\)
\(1-\dfrac{2}{3}=\dfrac{3}{3}-\dfrac{2}{3}=\dfrac{1}{3}\)
Replace the original expression with its equivalent simplified expression. Keep the pair of parentheses if the result is negative, is a fraction, or a multiplication is implied (a number before a left parenthesis).
\(2^3+3^2-5\left(\dfrac{14}{3}\right)-\left(\dfrac{3}{4}\right)^3+12\left(\dfrac{1}{3}\right)\)
Step 2: Power (exponent).
\(\boxed {2^3}+\boxed {3^2}-5\left(\dfrac{14}{3}\right)-\boxed {\left(\dfrac{3}{4}\right)^3}+12\left(\dfrac{1}{3}\right)\)
Operations required in step 2:
\(2^3=2×2×2=8\)
\(3^2=3×3=9\)
\(\left(\dfrac{3}{4}\right)^3=\dfrac{3}{4}×\dfrac{3}{4}×\dfrac{3}{4}=\dfrac{3×3×3}{4×4×4}=\dfrac{27}{64}\)
Replace all power notations with their equivalent simplified expression. If the result of the power is negative, enclose the result in a new pair of parentheses to avoid confusion with subtraction of negative numbers.
\(8+9-5\left(\dfrac{14}{3}\right)-\left(\dfrac{27}{64}\right)+12\left(\dfrac{1}{3}\right)\)
Step 3: Multiplication or Division.
Perform all required multiplication and division from left to right. When a number is placed next to a left parenthesis, a multiplication operation is implied. For example:
\(2(-3)\) means \(2×(-3)\)
\(5\left(\dfrac{14}{3}\right)\) means \(5×\left(\dfrac{14}{3}\right)\)
To avoid confusion, treat \(-2(3)\) as subtract \(2×3\) rather than add \((-2)×3\)
\(8+9-\boxed{5\left(\dfrac{14}{3}\right)}-\left(\dfrac{27}{64}\right)+\boxed{12\left(\dfrac{1}{3}\right)}\)
Operations required in step 3:
\(5\left(\dfrac{14}{3}\right)=5×\left(\dfrac{14}{3}\right)=\dfrac{5}{1}×\dfrac{14}{3}=\dfrac{5×14}{1×3}=\dfrac{70}{3}\)
\(12\left(\dfrac{1}{3}\right)=12×\left(\dfrac{1}{3}\right)=\dfrac{12}{1}×\dfrac{1}{3}=\dfrac{12×1}{1×3}=\dfrac{12}{3}=4\)
Replace the expression with the result of all multiplication operations. Remember to keep all signs intact and keep fractions in improper form rather than mixed number. Remove all unnecessary parentheses as following.
\(8+9-\dfrac{70}{3}-\dfrac{27}{64}+4\)
Rules of multiplication/division
Same sign produces a positive result.
Different sign produces a negative result. This rule applies to all constants (decimal, fraction, integer).
\(\leqalignno{
3×4=12\cr
(3)(4)=12\cr
3(4)=12\cr
3(-4)=-12\cr
-3(4)=-12\cr
-3(-4)=12\cr
(-3)(-4)=12}\)
Step 3: Addition or Subtraction.
Rules of addition/subtraction