Nested Parentheses in the Order of Operations

Like Russian dolls, some arithmetic expressions contain sets of nested parentheses — one set of parentheses inside another set. To evaluate a set of nested parentheses, start by evaluating the inner set of parentheses and work your way outward.

Parentheses — ( ) — come in a number of styles, including brackets — [ ] — and braces — { }. These different styles help you keep track of where a statement in parentheses begins and ends. No matter what they look like, to the mathematician these different styles are all parentheses, so they all get treated the same.

Sample question

  1. Find the value of {3 x [10 / (6 – 4)]} + 2.

    17. Begin by evaluating what’s inside the innermost set of parentheses: 6 – 4 = 2:

    {3 x [10 / (6 – 4)]} + 2 = {3 x [10 / 2]} + 2

    The result is an expression with one set of parentheses inside another set, so evaluate what’s inside the inner set: 10 / 2 = 5:

    = {3 x 5} + 2

    Now, evaluate what’s inside the final set of parentheses:

    = 15 + 2

    Finish up by evaluating the addition: 15 + 2 = 17.

Practice questions

  1. Evaluate 7 + {[(10 – 6) x 5] + 13}.

  2. Find the value of [(2 + 3) – (30 / 6)] + (–1 + 7 x 6).

  3. –4 + {[–9 x (5 – 8)] / 3} = ?

  4. Evaluate {(4 – 6) x [18 / (12 – 3 x 2)]} – (–5).

Following are the answers to the practice questions:

  1. 7 + {[(10 – 6) x 5] + 13} = 40. First evaluate the inner set of parentheses:

    7 + {[(10 – 6) x 5] + 13} = 7 + {[4 x 5] + 13}

    Move outward to the next set of parentheses:

    = 7 + {20 + 13}

    Next, handle the remaining set of parentheses:

    = 7 + 33

    To finish, evaluate the addition:

    7 + 33 = 40

  2. [(2 + 3) – (30 / 6)] + (–1 + 7 x 6) = 41. Start by focusing on the first set of parentheses. This set contains two inner sets of parentheses, so evaluate these two sets from left to right:

    [(2 + 3) – (30 / 6)] + (–1 + 7 x 6)

    = [(5) – (30 / 6)] + (–1 + 7 x 6)

    = [5 – 5] + (–1 + 7 x 6)

    Now, the expression has two separate sets of parentheses, so evaluate the first set:

    = 0 + (–1 + 7 x 6)

    Handle the remaining set of parentheses, evaluating the multiplication first and then the addition:

    = 0 + (–1 + 42) = 0 + 41

    To finish, evaluate the addition:

    0 + 41 = 41

  3. –4 + {[–9 x (5 – 8)] / 3} = 5. Start by evaluating the inner set of parentheses:

    –4 + {[–9 x (5 – 8)] / 3} = –4 + {[–9 x –3)] / 3}

    Move outward to the next set of parentheses:

    = –4 + [27 / 3]

    Next, handle the remaining set of parentheses:

    = –4 + 9

    Finally, evaluate the addition:

    –4 + 9 = 5

  4. {(4 – 6) x [18 / (12 – 3 x 2)]} – (–5) = –1. Focus on the inner set of parentheses, (12 – 3 x 2). Evaluate the multiplication first and then the subtraction:

    {(4 – 6) x [18 / (12 – 3 x 2)]} – (–5)

    = {(4 – 6) x [18 / (12 – 6)]} – (–5)

    = {(4 – 6) x [18 / 6]} – (–5)

    Now the expression is an outer set of parentheses with two inner sets. Evaluate these two inner sets of parentheses from left to right:

    = {–2 x [18 / 6]} – (–5) = {–2 x 3} – (–5)

    Next, evaluate the final set of parentheses:

    = –6 – (–5)

    Finish by evaluating the subtraction:

    –6 – (–5) = –1

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