Standard order of operations
1. Parentheses and fraction bars†Use the standard order of operations shown here to calculate the values of all expressions inside parentheses or brackets first, working from the innermost parentheses out. When dealing with a fraction bar, think of the entire numerator and denominator as being enclosed in parentheses, so calculate the numerator and denominator separately.
2. ExponentsRaise all numbers to the indicated powers.
3. Multiplication and divisionDo all the multiplications and divisions from left to right. Note on division: When division of integers leads to a fraction, it is often best to leave the fraction in reduced form rather than approximating by a decimal. (So, sometimes there is no calculation to do, as in $2/3,$ for instance.)
4. Addition and subtraction Do the remaining additions and subtractions from left to right.
†
Fraction bars are the horizontal lines separating the numerator and denominator in a fraction, as in $\dfrac{3-4}{6}$. The division signs $\div$ and $/$ do not count as fraction bars.
Remembering the order of operations: PEMDAS
| P | | Parentheses and fraction bars |
| E | | Exponents |
| MD | | Multiplication and Division (from left to right) |
| AS | | Addition and Subtraction (from left to right) |
Examples:
1. To calculate $6/3+4-1$:
- Parentheses and fraction bars: There are none in this example.
- Exponents: There are none in this example.
- Multiplication and division: Do the single division $6/3$:
$\color{indianred}{6/3} + 4 - 1 = 2 + 4-1$.
- Addition and subtraction: Do the remaining additions and subtractions from left to right:
$\color{indianred}{2+4} - 1 = 6 - 1 = 5$.
2. To calculate $6/(4-6) \times 2 -2^3$:
- Parentheses and fraction bars: First calculate the value of the entire quantity in parentheses (using the the order of operations as necessary).
$\color{indianred}{6/(4-6)} \times 2 -2^3 = 6/(-2) \times 2 -2^3$.
- Exponents: Raise all numbers to the indicated powers.
$6/(-2) \times 2 -\color{indianred}{2^3} = 6/(-2) \times 2 - 8$.
- Multiplication and division: Do all the multiplications and divisions from left to right:
$\color{indianred}{6/(-2)} \times 2 - 8 =\color{indianred}{-3 \times 2} - 8 = -6 - 8$.
- Addition and subtraction: Do the remaining additions and subtractions from left to right:
$\color{indianred}{-6 - 8} = -14$.
Some for you
In each of the following, decide which calculation of the given expression is valid.
Note If an answer happens to be correct, it does not necesssarily mean that the calculation is valid.
Do not use a calculator; using a calculator would completely defeat the purpose of these exercises.