If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Lesson 5: More on order of operations

# Exponents and order of operations FAQ

## What are exponents?

Exponents are a way of writing repeated multiplication. For example, instead of writing 2, times, 2, times, 2, times, 2, we can write 2, start superscript, 4, end superscript. The number 2 is called the base, and the number 4 is called the exponent. The exponent tells us how many factors of the base to include in the product. So, 2, start superscript, 4, end superscript means 2, times, 2, times, 2, times, 2, equals, 16. Exponents are useful when we want to write big numbers or compare numbers that have a lot of factors.

## Why do we need exponents?

Mathematicians who wanted to make their work easier and faster invented exponents. Imagine if you had to write or say a number like 1, comma, 000, comma, 000, comma, 000, comma, 000. That would take a lot of time and space, right? But with exponents, we can write it as 10, start superscript, 12, end superscript. That means:
start underbrace, 10, times, 10, times, 10, times, 10, times, 10, times, 10, times, 10, times, 10, times, 10, times, 10, times, 10, times, 10, end underbrace, start subscript, 12, start text, space, f, a, c, t, o, r, s, end text, end subscript
See how much shorter and simpler using the exponent is?
We use exponents to write and compare very large or very small numbers, like the size of the universe, the speed of light, the population of the world, or the weight of an atom.
We also use exponents to model and predict patterns of growth or decay, like the interest on a bank account, the spread of a virus, the life span of a battery, or the half-life of a radioactive substance.

## How do we find powers of numbers?

To find the value of a power of a whole number, we multiply the base by itself as many times as the exponent. Here's an example.
\begin{aligned} 3^2 &= \underbrace{3 \times 3}_{2\text{ factors}} \\\\ &= 9 \end{aligned}
Powers of fractions and decimals are just exponents with fractions and decimals as bases. To find the value of a power of a fraction or a decimal, we multiply the base by itself as many times as the exponent. Here's an example:
\begin{aligned} \left(\dfrac{5}{2}\right)^3 &= \dfrac{5}{2} \times \dfrac{5}{2} \times \dfrac{5}{2} \\\\ &= \dfrac{125}{8} \end{aligned}

## Where does the order of operations come from?

The order of operations, or the convention of performing arithmetic calculations in a certain sequence, is not a universal or natural law, but a human invention that developed over time and across cultures. Different peoples had their own ways of organizing and expressing mathematical operations, depending on their needs, preferences, and traditions.
For example, in ancient India, mathematicians used a system of symbols and rules called the siddhanta, which included the concepts of zero, negative numbers, fractions, algebra, and trigonometry. They followed a general order of operations that was similar to the modern one, except that they gave exponentiation the highest priority, followed by roots, then multiplication and division, and finally addition and subtraction.
In ancient China, mathematicians used a system of rods or characters to represent numbers and operations, and a counting board or an abacus to perform calculations. They also had a general order of operations that was similar to the modern one, except that they gave multiplication and division the same priority as addition and subtraction, and used parentheses to indicate the order of nested expressions.
In ancient Egypt, mathematicians used a system of hieroglyphs and fractions to represent numbers and operations, and a papyrus or a slate to perform calculations. They did not have a fixed order of operations, but rather relied on the context and the layout of the problem to determine the sequence of steps. They often used unit fractions, or fractions with a numerator of one, to simplify complex fractions, and used the method of false position to solve equations.
The history of the order of operations used in the United States can be traced back to the 16th and 17th centuries, when mathematicians such as Francois Viete, Rene Descartes, and Gottfried Leibniz developed the modern algebraic notation and the rules for manipulating powers and roots. They also introduced the use of parentheses to indicate grouping and precedence of operations. However, there was no universal agreement on the order of multiplication and division, or addition and subtraction, until the 19th century.
The first explicit mention of the order of operations in a textbook was in 1917, by David Eugene Smith and William David Reeve in their book A First Course in Algebra. They used the term "hierarchy of operations" and stated that "operations inclosed [sic] in parentheses are to be performed before any others; then, of the remaining operations, those indicated by exponents, radicals, or vincula are to be performed before multiplication or division, and these before addition or subtraction." They also used the term "vinculum" to refer to a horizontal line that groups terms, such as start overline, a, plus, b, end overline.
The acronym PEMDAS was popularized by William Betz in his 1958 book Arithmetic, A Modern Approach. He used the phrase "please excuse my dear Aunt Sally" as a mnemonic device to help students remember the order. However, some variations of the acronym exist, such as GEMA (grouping, exponents, multiplication and division, addition and subtraction) or BODMAS (brackets, orders, division and multiplication, addition and subtraction).

## When do we NOT follow the order of operations?

Lots of times, actually! While the order of operations gives us one way to evaluate an expression, the properties of addition and multiplication allow us to be more flexible.
• The distributive property says that we can multiply a value to each term inside of the parentheses instead of adding or subtracting inside the parentheses first.
• The commutative property of addition says that we can add the terms in any order instead of only left to right. That will become really powerful once we learn more about negative numbers, because then we'll learn a way to rewrite expressions with addition in place of the subtraction.
• The commutative property of multiplication says that we can multiply the factors in any order instead of only left to right. Once we learn more about reciprocals, we'll be able to rewrite expressions with multiplication in place of the division.