Order of Operations Calculator with Steps

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expressions with parentheses, brackets and braces

Expressions with parentheses, brackets and braces

Paste the expression into the field and every step appears with the reduced part highlighted. Grouping symbols say who goes first; the tool solves every innermost group at once, as a textbook does, and only then works outward.

The order of the groups and what happens inside each one

Brazilian textbooks use three symbols for one idea: parentheses ( ) at the innermost level, brackets [ ] wrapping parentheses and braces wrapping brackets. The rule is to solve what is in parentheses first, then brackets and last the braces. That is the same as always going from the innermost group outward. Three symbols only make the nesting easier to see; the arithmetic would be identical with parentheses at every level.
Inside each group the order of operations applies: powers and roots, then multiplications and divisions left to right and, last, additions and subtractions also left to right. In {[(8 − 3) × 2 + 2²] ÷ 7} + 3 the innermost group is (8 − 3) = 5. In the bracket what is left is 5 × 2 + 2²: first the power 2² = 4, then 5 × 2 = 10 and then 10 + 4 = 14. In the braces, 14 ÷ 7 = 2, and outside them 2 + 3 = 5. That is 6 steps, and the tool shows all 6.
Three mistakes keep appearing. The first is solving from the outside in, which gives the wrong result: in 100 − {20 + [5 × (2 + 3) − 10]} the right path reaches 100 − 35 = 65. The second is forgetting that a minus sign before a group applies to the whole group: 100 − (20 + 15) is 65, not 100 − 20 + 15 = 95. The third is dropping a closing symbol; the tool reports which opening symbol has no partner and at what position.

Checked examples

Parentheses only

Input: (8 + 4) × 3 − 5
Expected output: 31

12 × 3 = 36 and 36 − 5 = 31.

Brackets over parentheses

Input: [(6 + 2) × 3 − 4] ÷ 5
Expected output: 4

8 × 3 = 24, 24 − 4 = 20, 20 ÷ 5 = 4.

All three levels, with a power

Input: {[(8 − 3) × 2 + 2²] ÷ 7} + 3
Expected output: 5

6 steps, from the subtraction to the final sum.

Subtraction inside braces

Input: {15 − [4 + (6 − 2) × 2]} ÷ 3
Expected output: 1

4 + 8 = 12 and 15 − 12 = 3.

Minus sign before the bracket

Input: 12 − [3 + (8 − 5)]
Expected output: 6

The bracket is worth 6; 12 − 6 = 6.

Braces with several operations

Input: 100 − {20 + [5 × (2 + 3) − 10]}
Expected output: 65

25 − 10 = 15; 20 + 15 = 35; 100 − 35.

Full tool FAQ

What is the order for solving a numeric expression?

Groups first (parentheses, then brackets, then braces, always innermost outward). Inside each group: powers and roots; multiplications and divisions, left to right; additions and subtractions, left to right. Example: 2 + 3 × 4² = 2 + 3 × 16 = 2 + 48 = 50.

Why are there parentheses, brackets and braces?

Does multiplication always come before division?

What is the difference between −3² and (−3)²?

How does it handle fractions and mixed numbers?

How do the sign rules work?

What about 6 ÷ 2(1 + 2), the viral expression?

What happens with roots that are not exact?

Which symbols does the tool accept?

When does the tool show an error?

Frequently asked questions

Do I have to use all three kinds of symbol?

No. Any combination works as long as each closing symbol matches its opening one. The parentheses, brackets, braces convention only helps reading. With more than three levels, repeat the symbols.

Are two groups at the same level solved together?

Yes, they are independent. In (2 + 3) × (4 − 1) the tool solves both parentheses in the same step and reaches 5 × 3. That saves steps without changing the result.

What does a number touching a parenthesis mean, like 2(3 + 4)?

It is a multiplication with no sign: 2(3 + 4) = 2 × 7 = 14. The tool inserts the times sign and solves the parentheses first, as always.

How do I type braces and brackets on a phone?

Use the symbol keyboard under the field: the buttons insert ( ) [ ] and braces at the cursor, along with times, divide, power and root.

Next steps

The order of the groups and what happens inside each one

Brazilian textbooks use three symbols for one idea: parentheses ( ) at the innermost level, brackets [ ] wrapping parentheses and braces wrapping brackets. The rule is to solve what is in parentheses first, then brackets and last the braces. That is the same as always going from the innermost group outward. Three symbols only make the nesting easier to see; the arithmetic would be identical with parentheses at every level.

Inside each group the order of operations applies: powers and roots, then multiplications and divisions left to right and, last, additions and subtractions also left to right. In {[(8 − 3) × 2 + 2²] ÷ 7} + 3 the innermost group is (8 − 3) = 5. In the bracket what is left is 5 × 2 + 2²: first the power 2² = 4, then 5 × 2 = 10 and then 10 + 4 = 14. In the braces, 14 ÷ 7 = 2, and outside them 2 + 3 = 5. That is 6 steps, and the tool shows all 6.

Three mistakes keep appearing. The first is solving from the outside in, which gives the wrong result: in 100 − {20 + [5 × (2 + 3) − 10]} the right path reaches 100 − 35 = 65. The second is forgetting that a minus sign before a group applies to the whole group: 100 − (20 + 15) is 65, not 100 − 20 + 15 = 95. The third is dropping a closing symbol; the tool reports which opening symbol has no partner and at what position.

Checked examples

Parentheses only

Input: (8 + 4) × 3 − 5
Expected output: 31

12 × 3 = 36 and 36 − 5 = 31.

Brackets over parentheses

Input: [(6 + 2) × 3 − 4] ÷ 5
Expected output: 4

8 × 3 = 24, 24 − 4 = 20, 20 ÷ 5 = 4.

All three levels, with a power

Input: {[(8 − 3) × 2 + 2²] ÷ 7} + 3
Expected output: 5

6 steps, from the subtraction to the final sum.

Subtraction inside braces

Input: {15 − [4 + (6 − 2) × 2]} ÷ 3
Expected output: 1

4 + 8 = 12 and 15 − 12 = 3.

Minus sign before the bracket

Input: 12 − [3 + (8 − 5)]
Expected output: 6

The bracket is worth 6; 12 − 6 = 6.

Braces with several operations

Input: 100 − {20 + [5 × (2 + 3) − 10]}
Expected output: 65

25 − 10 = 15; 20 + 15 = 35; 100 − 35.

Full tool FAQ

What is the order for solving a numeric expression?

Why are there parentheses, brackets and braces?

Does multiplication always come before division?

What is the difference between −3² and (−3)²?

How does it handle fractions and mixed numbers?

How do the sign rules work?

What about 6 ÷ 2(1 + 2), the viral expression?

What happens with roots that are not exact?

Which symbols does the tool accept?

When does the tool show an error?

Frequently asked questions

Do I have to use all three kinds of symbol?

No. Any combination works as long as each closing symbol matches its opening one. The parentheses, brackets, braces convention only helps reading. With more than three levels, repeat the symbols.

Are two groups at the same level solved together?

Yes, they are independent. In (2 + 3) × (4 − 1) the tool solves both parentheses in the same step and reaches 5 × 3. That saves steps without changing the result.

What does a number touching a parenthesis mean, like 2(3 + 4)?

It is a multiplication with no sign: 2(3 + 4) = 2 × 7 = 14. The tool inserts the times sign and solves the parentheses first, as always.

How do I type braces and brackets on a phone?

Use the symbol keyboard under the field: the buttons insert ( ) [ ] and braces at the cursor, along with times, divide, power and root.