Order of Operations
Every question with a fully worked solution.
- Order of Operations - Video - Order of Operations Watch
Theory
Work out grouping symbols first (brackets, the fraction bar and the square root sign), then powers and roots, then multiply and divide from left to right, and add and subtract last, from left to right. This Year 8 maths page uses integers throughout.
1. Simplify inside grouping symbols, starting with the innermost brackets. 2. Work out powers and roots. 3. Multiply and divide from left to right. 4. Add and subtract from left to right.
Three things group: brackets, the fraction bar and the square root sign. Brackets inside brackets are worked from the inside out; the outer pair is often written \([\;]\).
A number against a bracket means multiply: \(-4(2 - 9) = -4 \times (-7) = 28\). The root sign groups everything under it: \(\sqrt{40 + 9} = \sqrt{49} = 7\).
Brackets change the value: \(30 - 4 \times 5 = 10\), but \((30 - 4) \times 5 = 130\).
Substituting. Put each negative value in brackets as you substitute, so \(a^2\) with \(a = -6\) becomes \((-6)^2 = 36\), not \(-6^2\). \(ab\) means \(a \times b\).
Calculators. A calculator does not see a fraction bar. To find a mean, type the brackets: \((-14 + 22 - 5 + 9) \div 4 = 3\). Without them it divides only the last number.
Missing number. Undo the operations in reverse order, then check by working forwards.
The order of operations:
| Step | What to do |
|---|---|
| 1 | Grouping symbols: brackets (inside out), fraction bar, square root sign |
| 2 | Powers and roots |
| 3 | \(\times\) and \(\div\), from left to right |
| 4 | \(+\) and \(-\), from left to right |
How to evaluate an expression
- Work out inside grouping symbols, innermost first.
- Work out powers and roots.
- Multiply and divide from left to right.
- Add and subtract from left to right.
How to work backwards
- List the operations in the order they are done.
- Undo them in reverse order.
- Check by substituting your answer.
\(17 - (4 + 3) \times (-2)\)
| \(\) | \(=\) | \(17 - 7 \times (-2)\) |
| \(\) | \(=\) | \(17 - (-14)\) |
| \(\) | \(=\) | \(17 + 14 = 31\) |
\(31\)
\((4 - (2 + 1)) + 12 \div 3\)
| \(\) | \(=\) | \((4 - 3) + 12 \div 3\) |
| \(\) | \(=\) | \(1 + 12 \div 3\) |
| \(\) | \(=\) | \(1 + 4 = 5\) |
\(5\)
| \((-3)^2\) | \(=\) | \(9\) |
| \(5 \times (-4)\) | \(=\) | \(-20\) |
| \(7 - (-20) + 9\) | \(=\) | \(7 + 20 + 9 = 36\) |
\(36\)
(i) \(-6[5 - (2 - 6)]\)
(ii) \(\sqrt{100 - 36} \times (-5)\)
| \(5 - (2 - 6)\) | \(=\) | \(5 - (-4) = 9\) |
| \(-6 \times 9\) | \(=\) | \(-54\) |
| \(\sqrt{100 - 36}\) | \(=\) | \(\sqrt{64} = 8\) |
| \(8 \times (-5)\) | \(=\) | \(-40\) |
(i) \(-54\)
(ii) \(-40\)
(i) \(a^2 + ab\)
(ii) \(\dfrac{ab}{c} - a\)
| \((-6)^2 + (-6) \times 5\) | \(=\) | \(36 + (-30)\) |
| \(\) | \(=\) | \(6\) |
| \(\dfrac{(-6) \times 5}{-3} - (-6)\) | \(=\) | \(10 + 6 = 16\) |
(i) \(6\)
(ii) \(16\)
Undo \(-3\): the fraction is \(5 + 3 = 8\). Undo \(\div (-4)\): \({?} + 6 = 8 \times (-4) = -32\). Undo \(+6\): \({?} = -32 - 6 = -38\).
| \(\dfrac{-38 + 6}{-4} - 3\) | \(=\) | \(\dfrac{-32}{-4} - 3\) |
| \(\) | \(=\) | \(8 - 3 = 5\) |
\({?} = -38\)
(ii) Ben wants the mean of \(-14\), \(22\), \(-5\) and \(9\). He types \(-14 + 22 - 5 + 9 \div 4\). What does the calculator show, and what is the mean?
| \(\text{final}\) | \(=\) | \(-6 + 3 \times 4 - 2 \times 3\) |
| \(\) | \(=\) | \(-6 + 12 - 6 = 0\) |
| \(\text{shown}\) | \(=\) | \(3 + 2.25 = 5.25\) |
| \(\text{mean}\) | \(=\) | \(12 \div 4 = 3\) |
(i) \(0^\circ\)C
(ii) It shows \(5.25\), because it divides only the \(9\). The mean is \(3\).
Common pitfalls
Frequently asked questions
What is the order of operations?
Grouping symbols, then powers and roots, then multiply and divide from left to right, then add and subtract from left to right.
Do you multiply before you divide?
No. They have equal priority, so work from left to right: \(12 \div 3 \times 2 = 8\).
Is the fraction bar a grouping symbol?
Yes. Work out the whole top and the whole bottom before dividing.
Why do I put a negative number in brackets when I substitute?
So that the whole number is squared: \((-6)^2 = 36\), but \(-6^2 = -36\).
How do I find a missing number in a calculation?
Undo the operations in reverse order, then check by working forwards.