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Introduction to Roots
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Introduction

In this lesson, you will learn about square roots. Just as multiplication and division are opposites of one another, powers and roots are opposite.

Real World Application

Some calculators have you type the number in first and then hit the square root button. Other calculators may have you do it the opposite way, by selecting the square root button and then typing in the number you want to root. You should experiment with your calculator on a simple square root, such as \(\sqrt{9}=3\), in order to see how your calculator works.


This video illustrates the lesson material below. Watching the video is optional.


Introduction to Roots

The word root has different meanings. If talking about a plant or a tree, the roots are the little parts of the plant that go down into the soil, and they provide nourishment. In math, however, a root is something like this:

\begin{align*} ROOT: \sqrt{\;\;\;\;\;}\end{align*}

These two types of roots have something in common: they are both the source of something. The roots on a plant are the source of the nourishment. A root in math literally means the source of a particular number.

Example 1
You know from the multiplication facts that \(3\times3=9\). Suppose you didn’t know that, and you just had the number 9. You want to know what one number you can multiply together twice or what multiplied by itself will equal 9. You express that question in math by using a symbol that represents the second root. The symbol, called a radical, is shown below:

\begin{align*} \sqrt{\;9\;} \end{align*}

This symbol asks the question, “What multiplied by itself will equal \(9\)?” Based on the multiplication facts, the answer is \(3\).
\begin{align*} \sqrt{\;9\;} = 3 \end{align*}
The second root, or the two root, is often called a square root. You usually use a radical alone as the square root symbol, but you could also put a \(2\) next to it to confirm that you are finding the square root: \(\sqrt[2]{\;9\;}\).

Example 2
What is the square root of 81?

\begin{align*} \sqrt{\;81\;} \end{align*}

If you know the multiplication facts, square roots are easy to figure out. You know that \(9\times9=81\), so

\begin{align*} \sqrt{\;81\;} = 9 \end{align*}

Using a Calculator


Suppose you don’t know the multiplication facts, but you still need to find the square root of 81. You can use a calculator. There may be a button like this on your calculator with a little radical sign: \(\sqrt[2]{\;x\;}\), \(\sqrt[3]{\;x\;}\), \(\sqrt[y]{\;x\;}\)

The 2 next to the radical tells you that it’s the square root.

Here’s an example of an online calculator whose radical sign looks like a little check mark. If a radical button doesn’t have an index number, it represents a square root.

This figure shows a calculator with a radical sign that looks like a check mark.

Figure 1

When calculating a square root, the calculator may require you to press the radical button either before or after you type in the number. Please check your calculator and discover which option you will need to use.

It’s also important to remember that roots and squares are opposites. That means you can use a root to reverse a square, and you can use a square to reverse a root. \(2^2= 4\) and \(\sqrt{\;4\;} = 2\) are simple examples to follow with these reverse patterns.

Example 3
Suppose you need to find the square root of a number that isn’t a perfect square. A perfect square is a number whose square root is a whole number. These numbers include 4, 9, 16, 25, 36, 49, and so on, because \(2\times2=4\), \(3\times3=9\), \(4\times4=16\), and so forth.

12 is not a perfect square therefore you cannot simplify the \(\sqrt{\;12\;}\) to be a whole number. You can try to estimate it by hand, but one of the advantages of a calculator is that it can perform these calculations for us.

In order to calculate the square root of 12 on this calculator, type in 12 and then press the square root button.

This figure shows how to calculate the square root of 12 by pressing the square root button.

Figure 2

Since the answer has lots of numbers after the decimal point, round to the nearest hundredth. The answer is 3.46. Test this answer by squaring it. \(3.46^{2}=11.9716\), which is close to 12.

This figure shows that you have to round the answer to the nearest hundredth.

Figure 3

The more digits after the decimal place you use, the more accurate the answer will be. If you squared 3.464 instead, you would get 11.999296 as the answer, which is closer to 12. Keep this in mind when performing multi-step equations.


Things to Remember


  • When calculating a square root, a calculator may require you to press the radical button either before or after you type in the number. 
  • Roots and squares are opposite operations.
  • If you use more decimal-place digits of a square root in your calculations, you will get a more accurate answer.

Practice Problems

Evaluate the following expressions:
  1. \(\sqrt{64}=\) (
    Solution
    x
    Solution: 8
    Details:
    The square root is looking for the number when multiplied to itself is equal to the number under the radical sign \(\sqrt{}\).

    It is highly suggested that you learn the multiplication facts. These will help you with questions like this one. According to our multiplication facts:

    \(8 \times 8 = 64\)

    Therefore, \(\sqrt{64} = 8\)
    )
  2. \(\sqrt{49}=\) (
    Solution
    x
    Solution: 7
    )
  3. \(\sqrt{121}=\) (
    Video Solution
    | Transcript)
  4. \(\sqrt{7}=\) (
    Solution
    x
    Solution: \(2.65\) (Rounded to the nearest hundredth)
    Details:
    The square root is looking for the number when multiplied by itself is equal to the number under the radical sign \(\sqrt{}\).

    When a number is a perfect square, or the product of a whole number squared like \(3\times3=9\), then it is easier to find the square root. However, a number like 7 is not a perfect square. In cases like this, you can estimate the square root. It must be somewhere between 2 and 3 because \(2\times2=4\) and \(3\times3=9\) and 7 is between 4 and 9. In order to be more accurate though, use a calculator.

    Your calculator may have a button that looks similar to one of these:

    This image shows three different types of square root you may have on a calculator. They all represent the same thing. First is \sqrt{x} with index 2, second is the square root symbol, and the last one is \sqrt{x}.

    Using a calculator, you should get the answer \(2.645751311\)…

    You can then round it to the precision needed for the question. In this case, you will round it to the nearest hundredth which is \(2.65\).
    )
  5. \(\sqrt{78}=\) (
    Video Solution
    x
    Solution: \(8.83\) (Rounded to the nearest hundredth)
    Details:

    (Introduction to Roots #5 (01:45 mins) | Transcript)
    | Transcript)

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