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H.3 Evaluate numerical expressions

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What does it mean to evaluate numerical expressions?

To evaluate a numerical expression means to find its value by performing the operations in the correct order. An expression can include addition, subtraction, multiplication, division, and grouping symbols such as parentheses, brackets, or braces.

Example:
  • Expression: 8 + 3 × 2
  • Step 1: Multiply first → 3 × 2 = 6
  • Step 2: Add next → 8 + 6 = 14
  • Value of the expression = 14
Note

Always follow the Order of Operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). You can remember this order with the phrase “Please Excuse My Dear Aunt Sally.”

Using parentheses ( ) in expressions

Parentheses show which operations should be done first in an expression. Anything inside parentheses must be solved before operations outside of them.

Example:
  • Expression: (6 + 4) × 3
  • Step 1: Solve inside parentheses → 6 + 4 = 10
  • Step 2: Multiply → 10 × 3 = 30
  • Value of the expression = 30
Note

If an expression has parentheses, complete all operations inside them first. This helps keep the expression clear and accurate.

Using brackets [ ] in expressions

Brackets are used when an expression already has parentheses. They help group parts of the expression and show which operations to do next after the parentheses.

Example:
  • Expression: [5 + (3 × 2)] × 4
  • Step 1: Solve inside parentheses → 3 × 2 = 6
  • Step 2: Solve inside brackets → 5 + 6 = 11
  • Step 3: Multiply → 11 × 4 = 44
  • Value of the expression = 44
Note

Work from the inside out: first solve parentheses, then brackets, and finally any operations outside the brackets.

Using braces { } in expressions

Braces are used for grouping larger or more complex expressions. They often contain brackets and parentheses inside them to organize many steps clearly.

Example:
  • Expression: {2 + [3 × (4 + 1)]} × 2
  • Step 1: Solve inside parentheses → 4 + 1 = 5
  • Step 2: Solve inside brackets → 3 × 5 = 15
  • Step 3: Solve inside braces → 2 + 15 = 17
  • Step 4: Multiply → 17 × 2 = 34
  • Value of the expression = 34
Note

When an expression has braces, brackets, and parentheses, always start with the innermost grouping and move outward step by step.

Checking your work

After evaluating an expression, check each step to make sure you followed the correct order of operations and performed every operation accurately.

Example:
  • Expression: (8 ÷ 2) + (3 × 4)
  • Step 1: Inside first parentheses → 8 ÷ 2 = 4
  • Step 2: Inside second parentheses → 3 × 4 = 12
  • Step 3: Add results → 4 + 12 = 16
  • Value of the expression = 16
Note

If you get a different answer when checking, look back at the grouping symbols and make sure you followed the order correctly. Careful steps lead to accurate results.