Simplifying Negative Square Roots

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Lesson Priority: High

Algebra Two $\longrightarrow$
Complex Numbers $\longrightarrow$

Objectives
• Learn how to simplify square roots of negative numbers
• Apply what we already know about reducing square roots of positive integers
Lesson Description

Now that we understand what the imaginary number is and how it works, we can simplify the square root of negative numbers in a very similar way to how we simplify square roots of positive ones.

Practice Problems

Practice problems and worksheet coming soon!

I Thought We Couldn't Root Negatives?

By far, the most common and useful applications of the imaginary number involve simplifying square root expressions that contain negative numbers. For better or worse, this process is virtually a carbon copy of the skill of simplifying positive root expressions - so if you aren't caught up on how to simplify integer square roots » then get with the program, and then read on.It's worth mentioning that we probably have had at least one teacher who has specifically told us that we cannot take the square root of a negative number. Whether or not you want to consider that a lie depends on your perspective - it's true that you cannot square root a negative number if you are only going to work with real numbers, but if you allow for the existence of imaginary numbers » , then you can actually square root anything (including square rooting imaginary numbers » , but that's not something we'll look at until after we know some trigonometry!). For all students taking Algebra 2 and beyond, it is necessary to work with imaginary numbers, at least at the basic level, and this lesson is the skill you absolutely need to know.Fortunately, as we said this is a straight-forward adaptation of what we already know about simplifying square roots. Specifically, here's how it works.
Theorem: Negative Square RootsSimplifying an expression involving the square root of a negative quantity is done by simplifying the quantity as if it was positive, and placing an $i$ in front as a coefficient. In other words,$$\sqrt{-x} = i\sqrt{x}$$For all $x$ such that $x$ is a positive number, so that $-x$ is a negative number.
Let's take a look at an example.

Example 1Simplify.$$\sqrt{-40}$$$\blacktriangleright$ Digging back to our knowledge of simplifying integer square roots, we realize that $\sqrt{40}$ is simplified as $2\sqrt{10}$. Therefore, applying Theorem 1, we have$$\sqrt{-40} = i\sqrt{40}$$$$= 2i\sqrt{10}$$

It's a simple adaptation to what we already know. We know $\sqrt{40} = 2\sqrt{10}$, so $\sqrt{-40} = 2i\sqrt{10}$.
Pro Tip
Sharpen up on your regular root simplification skills. Once you get in the groove on practicing this skill, you'll see that you're really just using that regular root skill that you already have, except you're simply sticking an $i$ in front of your answer. All the "hard work" revolves completely around the root simplification process you already know!
Here's another example.

Example 2Simplify.$$\sqrt{-72}$$$\blacktriangleright$ Via the square root simplification process we already know:$$\sqrt{72} = \sqrt{(36)(2)}$$$$=6 \sqrt{2}$$Once again, Theorem 1 above tells us that$$\sqrt{-72} = i \sqrt{72}$$It follows that$$\sqrt{-72} = i \cdot 6\sqrt{2}$$$$6i \sqrt{2}$$
Warning!
The Root Product Property fails for square roots of products of negative numbers due to the properties of the imaginary number!!! Be careful not to apply it verbatim! For example, let's try and apply the Root Product Property to the following example:$$\sqrt{(-10)(-7)}$$According to the Root Product Property:$$\sqrt{(-10)(-7)} = \sqrt{(-10)} \cdot \sqrt{(-7)}$$$$\longrightarrow = i\sqrt{10} \cdot i \sqrt{7} = i^2 \sqrt{10}\sqrt{7}$$$$= -\sqrt{10} \sqrt{7} = -\sqrt{70}$$But if we instead multiplied under the root first, then simplified, we would have got a completely different answer:$$\sqrt{(-10)(-7)} = \sqrt{70}$$As you can see, with the first perspective, the imaginary numbers created a negative sign outside of the radical. However, under the second perspective, the two negatives multiplied to a positive result before we even started trying to take square roots. It is the latter case that is correct. It looks dumbfounding, but the good news is, teacher and professors rarely try to use this apparent slight of hand against you. It is incredibly rare that this mishap is actually presented to you in a situation where you're getting graded.
Overall, the takeaway of this lesson is that usually you will simply be given a square root expression with a negative sign to simplify. Accomplish this by applying what we already know about simplifying square roots to get the number part correct, and use the idea that we end up with the same result as we would in the case of a positive square root, but with an $i$ factored out.

Put It To The Test

Try a few problems out on your own. Remember that there's nothing new here from what you already should know about simplifying numeric roots, in terms of the numbers. If you're looking for practice problems about other things we do with $i$, check out the other lessons on Imaginary Numbers ».

Example 3Simplify.$$\sqrt{-100}$$
Show solution
$$\blacktriangleright \,\, \sqrt{-100} = i \sqrt{100}$$$$= 10i$$

Example 4Simplify.$$\sqrt{-75}$$
Show solution
$$\blacktriangleright \,\, \sqrt{-75} = i \sqrt{75}$$$$=i \sqrt{25 \cdot 3} = 5i \sqrt{3}$$

Example 5Simplify.$$\sqrt{-56}$$
Show solution
$$\blacktriangleright \,\, \sqrt{-56} = i \sqrt{56}$$$$=i \sqrt{4 \cdot 14} = 2i \sqrt{14}$$

Example 6Simplify.$$\sqrt{-91}$$
Show solution
$$\blacktriangleright \,\, \sqrt{-91} = i \sqrt{91}$$$91$ is prime, so there is nothing further to simplify.

Lesson Takeaways
• Be able to simplify square roots when the radical contains a negative number
• Although it is unlikely to appear on a test, understand why the Root Product Property fails for square roots of the product of negative numbers.
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Lesson Metrics

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You Should Know - Somewhat elective information that may give you a broader understanding.

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