Roots & the Order of Operations

Instructor: Michael Eckert

Michael has a Bachelor's in Environmental Chemistry and Integrative Science. He has extensive experience in working with college academic support services as an instructor of mathematics, physics, chemistry and biology.

As roots are a type of exponent, they are an important part of the order of operations. In covering roots, we will be reviewing both what makes up the order of operations and how to solve for roots.

Roots and the Order of Operations

Before we get started looking at roots in the order of operations, we should first review what makes up the order of operations.


PEMDAS is a short and easy way of remembering the order of operations as a whole. It is an acronym, or a word made from the first letters of other words. An acronym is a shorter way to communicate a longer idea. For example, IDK is an acronym for the phrase I Don't Know.

What does PEMDAS stand for, you may ask? It stands for the order of operations where Parentheses are first, Exponents are second, Multiplication and Division are third, followed by Addition and Subtraction fourth. If we are given a string of math operations involving any of these six things, we know what order to solve each operation (going from left to right).

To illustrate, let's look at this string of the following operations: (1 + 2) + 3 ∗ 4 / 5 - 6 + 7 / (82 + 9 ∗ 10). The order of operations says that we must

1) square the 8 in parentheses

2) multiply the 9 and 10 in the parentheses

3) add the 1 and 2 in parentheses, giving us 3 + 3 ∗ 4 / 5 - 6 + 7 / (64 + 90)

4) add the 64 and 90 in parentheses

5) multiply the 3 and 4, giving us 3 + 12 / 5 - 6 + 7 / 154

6) divide the 12 by the 5 to get: 3 + 2.40 - 6 + 7 / 154

7) divide the 7 by the 154 to get : 3 + 2.40 - 6 + .05

8) add and subtract the rest from left to right where 3 + 2.4 - 6 + .05 = -.55


Square Root

Let's look at the most basic root: a square root. A square root is just another way of expressing any number with a 1/2 exponent. Let's find the square root of 4, which is written as √4 or 4(1/2). Take note that the denominator in the fraction (1/2) in the exponent of 4 is 2. When solving for √4, we ask ourselves ''what 2 same numbers must we multiply together to get 4?'' We know that we must multiply 2 times 2 to get 4. So, we say that the square root of 4 is 2. Another way of writing √4 is:


The 2 outside of the root symbol tells us how many identical numbers (in this case 2), that when multiplied, give us the number on the inside of the root symbol. For instance, say we are looking for the square root of 100.


We must ask ourselves, what two identical numbers that when multiplied, give us 100? The answer is 10 and 10.


Cube Root

In addition to square roots, there are roots with a higher number on the outside of the root symbol. For instance, there may be a root with a 3 outside of the root symbol, called a cube root. For example, the cube root of 8 or 8(1/3) is expressed as


To solve for this, we ask ourselves what 3 identical numbers, that when multiplied, give us the number on the inside of the root symbol? Since 2∗2∗2 = 8, then


The cube root of 8 is 2. For the cubed root of 1000, 10∗10∗10 = 1000, so


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