When doing mathematical calculations, it is critically important to understand the order in which specific types of mathematical operations must be performed. You must also be able to determine which groupings of numbers and variables get evaluated in which order.
To start, note that groupings of numbers and/or variables are symbolized by a nesting of parentheses (), braces [], and curly brackets {}. It is generally true that authors will nest groups, when necessary, according to the convention { [ ( ) ] }. These groupings indicate that calculations are performed by the inside group first, and then moving outward.
The table below summarizes the order of operations within each grouping, completing all of the operations within a grouping before moving out to the next grouping. Below the table a number of examples will be given, followed by an opportunity for you to test your understanding of the order of operations.
| Order | Name | Symbol |
| First | exponents | \(a^b\) |
| Second* | multiplication or division | \(a\times b\) or \(a\div b\) |
| Third* | addition or subtraction | \(a+b\) or \(a-b\) |
(Note: If you are uncomfortable working with exponents, you should review the exponents tutorial.)
Regarding arithmetic symbols: There are several ways in which multiplication and division are represented.
For multiplication: When indicating that two numbers should be multiplied, \(\times\), * and \(\cdot\) and considered to be equivalent representations; for example, \(3 \times 2\), \(3 * 2\), and \(3\cdot 2\) all mean \(2+2+2\) while \(2 \times 3\), \(2 * 3\), and \(2\cdot 3\) all mean \(3+3\) (note that either order gives the same answer, a behavior referred to as the commutative property of multiplication). When variables or parentheses are involved, the multiplication symbol is generally left off, such as \(2\pi\), \(3f\), \(xy\), or \(2(3x + y)\). The corresponding expressions with the times symbol explicitly included would be \(2 \times \pi\), \(3 \times f\), \(x \times y\), and \(2 \times (3 \times x + y)\) respectively. (If you don’t feel comfortable working with variables you may want to study the variables tutorial.)
For division, the \(\div\) symbol is sometimes used within a line to indicate division, but most of the time the \(/\) symbol or a fraction is used instead (if necessary, review the fractions tutorial). For example \(3 \div 2\), \(3/2\), and \(\displaystyle \frac{3}{2}\) equal exactly the same thing, namely \(1.5\). The reason that the \(/\) symbol or a fraction is used almost universally is that it also demonstrates clearly that the quantity being represented is a ratio. Note that we will never use the long division symbol (\(\tiny\overline{)\hspace{2ex}}\)) that you were likely introduced to when you first learned how to divide. That symbol is easily confused with the square root symbol (\(\small\sqrt{\hspace{2ex}}\)).
In this tutorial we will go ahead and use \(\times\) and \(\div\) to represent multiplication and division as explicitly as possible, but they will be used rarely beyond this tutorial, favoring instead \(\cdot\) or no symbol for multiplication and a slash \(/\) or a fraction for division.
Example 1
Determine the result of the following expression: $$3 \times 5 – 2.$$
Since multiplication takes precedent over addition, the multiplication must be performed first and then the subtraction step: $$3 \times 5 – 2 = 15 – 2 = 13.$$
Note that if you do the subtraction step first you would get \(3 \times 3\) which would equal \(9\) which is incorrect. Clearly the order matters.
Example 2
Consider
$$4 \times 12 + 6 \div 2.$$
As in the previous example, carry out the multiplication and division operations first and then the addition operation, or
$$48 + 3 = 51.$$
Example 3
What about calculations involving exponents?
$$3^2 + 2 \times 8^2$$
According to our order of operations rules, the operation with exponents occurs first, then multiplications or divisions, followed by additions or subtractions:
$$9 + 2 \times 64 = 9 + 128 = 137.$$
Example 4
What happens where numbers are grouped? Recall that the order of operations involves completing the inner-most grouping first, and working your way out.
$$\left[3 + (2 + 5 \times 3)^2\right]^3.$$
Starting inside the pair of parentheses and working outward:
| \( \left[3 + (2 + 5 \times 3)^2\right]^3\) | \(=\) | \( \left[3 + (2 + 15)^2\right]^3\) |
| \(=\) | \( (3 + 17^2)^3\) | |
| \(=\) | \( (3 + 289)^3\) | |
| \(=\) | \( 291^3\) | |
| \(=\) | \( \mbox{24,642,171}\) | |
| \(=\) | \( 2.464\,2171 \times 10^6\). |
Notes:
- The last result is what your answer looks like in scientific notation. If you don’t feel comfortable with scientific notation you should work through the scientific notation tutorial.
- The switch from square brackets to parentheses in the second line is common when there aren’t any additional nested groupings. However, there is technically nothing wrong with keeping the square brackets.
Example 5
In this example we will look at what is involved if variables are part of the expression. If you are uncomfortable about working with variables, you should review the variables tutorial first.
Recall that variables are nothing more than symbolic placeholders for numbers to be substituted later. In this example, let \( m_1\) represent the mass of one star in mutual orbit around a second star of mass \( m_2\). \( P\) is the orbital period of the two stars around one another. We will use the mass of the Sun as the unit of mass, and years as the unit of period. In this example let \( m_1 = 2\), \( m_2 = 0.5\), and \( P = 5\). Substituting into the expression,
$$(m_1 + m_2)P^2$$
gives
$$(2 + 0.5)\times 5^2 = 2.5 \times 5^2 = 2.5 \times 25 = 62.5.$$
After the numbers are substituted into the appropriate variables, the calculation proceeds just like any of the previous examples.
Quiz Yourself
Now that you have more familiarity with working with the order of operations, you should try some of the problems below to solidify your understanding. By the way, computers and many calculators write scientific notation in “computer-speak,” where \(10\) raised to an integer (either positive or negative) replaces \(\times 10^\text{exponent}\) with \(\text{E(exponent)}\) or \(\text{e(exponent)}\); in other words, \(3.14159 \times 10^9\) is represented by \(3.14159\text{E}9\). If your calculator has the \(\Large{\hat{ }}\) symbol, then you would enter the number as \(3.14159 \times 10{\Large{\hat{ }}}9\).
- Evaluate the expression \( 5 + 3 \times 2 – 4\).
- Evaluate the expression \( (5 + 3)\times (2 – 4)\).
- Determine the value of the expression \( 5 + 3^4\times 2 – 4^3.\)
- Determine the value of the expression \( \left(5 + 3^4\right)\times \left(2 – 4\right)^3.\)
- Find the result of \( \left[\left(5 + 3^4\right)\times 2\right]^3 – 4^3.\)
- If \( x = 4\), \( y = 3\), and \( z = 2\), determine the value of $$\left\{[3x – (2y + 4)^4 – z]^2 – 25\right\}^3.$$
Answers
- \(7\)
- \(-16\)
- \(103\)
- \(-688\)
- \(\mbox{5088,384} = 5.088\,384\times10^6\)
- \(\mbox{994,014,233,010,684,125,421,875} \approx 9.940\,14\times10^{23}\)