Order of Operations with Integers - Sixth Grade Mathematics

Evaluating expressions with positive and negative integers requires synthesizing PEMDAS hierarchy rules with signed arithmetic principles, particularly when managing negative bases and powers.

The Hierarchy with Signed Numbers

The order of operations remains constant across all number systems:
1. Grouping symbols: Parentheses ( ), Brackets [ ], Braces { }, and Absolute Value bars | |.

2. Exponents and Powers: Evaluate powers from left to right.

3. Multiplication and Division: Co-equal priority from left to right.

4. Addition and Subtraction: Co-equal priority from left to right.

Evaluating Powers of Negative Numbers

One of the most frequent errors in secondary mathematics involves distinguishing between a negative base inside parentheses and a negative sign outside parentheses.

+-------------------------------------------------------------------------+
|                  NEGATIVE BASES VERSUS NEGATIVE OPERATORS               |
+-------------------+-----------------------+-----------------------------+
| Expression        | Expanded Meaning      | Final Value                 |
+-------------------+-----------------------+-----------------------------+
| (-3)^2            | Base is -3:           | +9 (POSITIVE!)              |
| (Inside parens)   | (-3) * (-3)           | (Negative * Negative = Pos) |
+-------------------+-----------------------+-----------------------------+
| -3^2              | Base is +3 ONLY:      | -9 (NEGATIVE!)              |
| (No parens)       | -(3 * 3)              | (Opposite of 3 squared)     |
+-------------------+-----------------------+-----------------------------+
| (-2)^3            | Base is -2:           | -8 (NEGATIVE!)              |
| (Odd exponent)    | (-2) * (-2) * (-2)    | (Odd count of negatives)    |
+-------------------+-----------------------+-----------------------------+
| (-2)^4            | Base is -2:           | +16 (POSITIVE!)             |
| (Even exponent)   | (-2)*(-2)*(-2)*(-2)   | (Even count of negatives)   |
+-------------------+-----------------------+-----------------------------+

The Exponent Parity Rule:
- Any negative base raised to an EVEN power results in a POSITIVE value: (-a)^even = +
- Any negative base raised to an ODD power results in a NEGATIVE value:   (-a)^odd  = -

Step-by-Step Multi-Operation Walkthroughs

Systematic line-by-line evaluation prevents sign oversights.

Evaluate: 14 - 3 * (-2)^3 + | -16 / 4 |

Step 1: Evaluate grouping and exponents:
        - Exponent: (-2)^3 = (-2) * (-2) * (-2) = -8.
        - Absolute value grouping: | -16 / 4 | = | -4 | = 4.
        Expression becomes: 14 - 3 * (-8) + 4

Step 2: Perform Multiplication:
        - 3 * (-8) = -24.
        Expression becomes: 14 - (-24) + 4

Step 3: Perform Subtraction and Addition from left to right:
        - Apply subtraction rule: 14 - (-24) = 14 + 24 = 38.
        - Add: 38 + 4 = 42.

Final Value: 42.

Managing Fraction Bars with Integers

When an expression is arranged across a horizontal fraction bar, evaluate the entire numerator and the entire denominator as independent grouped expressions before dividing.

Evaluate: [ (-5)^2 - 7 ] / [ (-2) * 3 ]

Step 1: Evaluate Numerator:
        (-5)^2 = 25.
        25 - 7 = 18.

Step 2: Evaluate Denominator:
        (-2) * 3 = -6.

Step 3: Perform Final Division:
        18 / (-6) = -3.

Exercises

Exercise 1. Evaluate each power: (a) (-4)^2; (b) -4^2; (c) (-5)^3; (d) (-1)^100.

Exercise 2. Evaluate the numerical expression: (-18) / 3 - 4 * (-5) + (-7). Show each step.

Exercise 3. Evaluate the expression with nested grouping symbols: 5 * [ (-14) - (-2)^4 ] / (-10).

Solutions

Solution to Exercise 1. For (a) (-4)^2: base is -4; (-4) * (-4) = +16. For (b) -4^2: base is 4; -(4 * 4) = -16. For (c) (-5)^3: base is -5; (-5) * (-5) * (-5) = 25 * (-5) = -125. For (d) (-1)^100: the exponent 100 is even; any negative raised to an even power is positive, so (-1)^100 = +1.

Solution to Exercise 2. Scan for grouping and exponents; none present. Next, perform division and multiplication from left to right: first divide (-18) / 3 = -6; then multiply 4 * (-5) = -20. Rewrite expression: -6 - (-20) + (-7). Perform subtraction: -6 - (-20) = -6 + 20 = 14. Finally, add: 14 + (-7) = 7. The final result is 7.

Solution to Exercise 3. Evaluate inside the square brackets first: [ (-14) - (-2)^4 ]. Evaluate the exponent: (-2)^4 = (-2) * (-2) * (-2) * (-2) = 16. The bracket becomes: [ -14 - 16 ] = -14 + (-16) = -30. The expression is now: 5 * [ -30 ] / (-10). Perform multiplication and division from left to right: 5 * (-30) = -150. Finally, divide: -150 / (-10) = +15. The final answer is 15.