Order of Operations with Rational Numbers

When working with rational numbers—which include positive and negative whole numbers, fractions, and decimals—it is crucial to follow the standard order of operations. This ensures that everyone who solves a math problem gets the same correct answer.

The Rules (PEMDAS)

To evaluate any expression correctly, follow the PEMDAS rule:

1. Parentheses: Solve operations inside grouping symbols first (like parentheses, brackets, or long fraction bars).
2. Exponents: Evaluate powers and roots.
3. Multiplication and Division: Perform these operations from left to right as they appear.
4. Addition and Subtraction: Perform these operations from left to right as they appear.

Step-by-Step Examples

Let's apply these rules to evaluate a few expressions involving rational numbers.

Example 1

Evaluate: $-2 + 3 \times (-4)$

• Step 1: There are no exponents or operations inside parentheses to solve first (the parentheses around the $-4$ just indicate a negative number). We start with Multiplication.
• Step 2: Multiply $3 \times (-4) = -12$.
• Step 3: Substitute this back into the expression: $-2 + (-12)$.
• Step 4: Add: $-2 + (-12) = -14$.

Answer: $-14$

Example 2

Evaluate: $(-3)^2 - 4 \times \left(-\frac{1}{2}\right)$

• Step 1: Evaluate the Exponent first. $(-3)^2 = (-3) \times (-3) = 9$.
  • Expression becomes: $9 - 4 \times \left(-\frac{1}{2}\right)$
• Step 2: Next is Multiplication. Multiply $4 \times \left(-\frac{1}{2}\right) = -2$.
  • Expression becomes: $9 - (-2)$
• Step 3: Finally, Subtract. Subtracting a negative is the same as adding a positive.
  • $9 + 2 = 11$

Answer: $11$

Example 3

Evaluate: $(2 - 5) \times (-3) + 1$

• Step 1: Solve inside the Parentheses first. $2 - 5 = -3$.
  • Expression becomes: $(-3) \times (-3) + 1$
• Step 2: Perform the Multiplication. $(-3) \times (-3) = 9$.
  • Expression becomes: $9 + 1$
• Step 3: Add.
  • $9 + 1 = 10$

Answer: $10$

Important Tips

• Negative Exponents: Pay close attention to parentheses. $(-3)^2$ means $(-3) \times (-3) = 9$, but $-3^2$ means $-(3 \times 3) = -9$.
• Double Negatives: Remember that subtracting a negative number acts like addition (e.g., $5 - (-3) = 5 + 3$).
• Fractions: When multiplying fractions, multiply straight across the numerators and denominators. When adding or subtracting them, always find a common denominator first.