# 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

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

Evaluate 7 + {[(10 – 6) x 5] + 13}.

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

–4 + {[–9 x (5 – 8)] / 3} = ?

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

Following are the answers to the practice questions:

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 + 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

–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 – 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