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
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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
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Evaluate 7 + {[(10 – 6) x 5] + 13}.
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Find the value of [(2 + 3) – (30 / 6)] + (–1 + 7 x 6).
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–4 + {[–9 x (5 – 8)] / 3} = ?
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Evaluate {(4 – 6) x [18 / (12 – 3 x 2)]} – (–5).
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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
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[(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
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–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
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{(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