Adding and Subtracting Root Expressions

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Objectives
  • Know how to simplify root expressions to see if they are "like terms"
  • Add and subtract "like term" root expressions similar to the way we add and subtract variables
Lesson Description

Whether you're working with numbers only or with variables in the mix, we can only ever add or subtract expressions that have the same root piece, akin to the concept of "like terms". Using what we recently learned about simplifying square roots, we can understand when root expressions can and cannot be combined together using addition or subtraction.

 
Practice Problems

Practice problems and worksheet coming soon!

 
You Should Know
This lesson covers square roots only. If you are looking for lessons on advanced roots of higher degrees, such as cube roots, you will find them in Algebra Two ».

Combining Roots

Combining like terms » is one of the most core tenets of Algebra. We're going to keep the same ideas in mind when it comes to roots, because the idea is really about counting. When we say $2x + 7x = 9x$, it's because we're asking "how many $x$'s do we have if we start with $2$ of them and add $7$ more?" It's no different than counting a total.So why do we even need a lesson on combining roots if it seems the same? It's because your teacher isn't going to give you a quiz that asks you to add $2\sqrt{3}$ and $7\sqrt{3}$ (which is expectedly equal to $9\sqrt{3}$), it's because you'll be asked to add something like $\sqrt{18}$ and $\sqrt{32}$ (which is absolutely not equal to $\sqrt{50}$).

How It Works

This is one of those things in math that is somewhat isolated and specific. When you're asked to perform arithmetic with square root expressions, you need to do things that you will probably only do when working with square root expressions. That's one of many reasons students generally dislike studying roots and radicals.In order to do this correctly, you need to follow the principle of like terms.
Like Terms for RootsRoot expressions can only be combined together with addition or subtraction when the two have identical expressions "under" the radical symbol.For example, $13\sqrt{6}$ and $10\sqrt{6}$ can be added or subtracted, but $\sqrt{11}$ and $\sqrt{22}$ cannot be added or subtracted together.
 
Example 1Simplify the following two arithmetic statements.$$\mathrm{a)} \;\;\;\; 13\sqrt{6} + 10\sqrt{6}$$$$\mathrm{b)} \;\;\;\; \sqrt{11} + \sqrt{22}$$$\blacktriangleright$ Since the first sum is two "like term" radicals, we can combine them.$$13\sqrt{6} + 10\sqrt{6} = 23\sqrt{6}$$The second one cannot be combined. The correct answer is to say "already simplified" or just re-write the expression, because $\sqrt{11} + \sqrt{22}$ cannot be written any more concisely.
 

The Usual Task

Usually, it will turn out that simplifying the radicals » on their own will allow us to combine them somehow. This is the common case, and the one to expect on quizzes and tests.
 
Example 2Simplify:$$\sqrt{18} + \sqrt{32}$$$\blacktriangleright$ We cannot dive right in an add these two roots together because they are not the same under the radical. However, let's simplify each.$$\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$$$$\sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}$$Now we can re-write $\sqrt{18} + \sqrt{32}$ as$$3\sqrt{2} + 4\sqrt{2}$$which we can combine, since the rule tells us two roots can be combined if they have the same stuff under the radical symbol.$$3\sqrt{2} + 4\sqrt{2} = 7\sqrt{2}$$
Remember!
Sometimes two root expressions can be simplified into objects that can combine together, and sometimes they can't. If they cannot, then they stay separate.
 

Variables in the Mix

Recently, we learned how to simplify radicals with variable expressions », which means we can take this "simplify before combining" approach not only with numeric square roots, but also with variable expression ones.The same principle applies: if you can simplify the expressions into a common radical form, then they can be combined. If you cannot, then they cannot.
 
Example 3Subtract.$$\sqrt{12x^5 y} - \sqrt{48xy^3}$$$\blacktriangleright$ Factor out all the perfect squares from each radical to remove them and simplify.$$\sqrt{4x^4 \cdot 3xy} - \sqrt{16y^2 \cdot 3xy}$$$$2x^2\sqrt{3xy} - 4y\sqrt{3xy}$$At this point, because each radical has the same thing under the hood, I would combine them with factoring by grouping.$$\left( 2x^2 - 4y \right) \sqrt{3xy}$$
 

Mr. Math Makes It Mean

Three or Four at a TimeOne way teachers may try to trip you up on quizzes is by including multiple roots at once. When they do, sometimes only some of them are compatible with one another, and sometimes one of the radicals cannot interact with any other radical in the problem, meaning the correct procedure is to let it sit there.
 
Example 4Simplify.$$\sqrt{90} - \sqrt{40} + \sqrt{12}$$$\blacktriangleright$ As before, simplify each radical and see what we're dealing with.$$\sqrt{90} = \sqrt{9 \cdot 10} = 3\sqrt{10}$$$$\sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10}$$$$\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}$$The first two roots can be combined together but not the third one.$$\sqrt{90} - \sqrt{40} + \sqrt{12}$$$$\longrightarrow 3\sqrt{10} - 2\sqrt{10} + 2\sqrt{3}$$$$=\sqrt{10} + 2\sqrt{3}$$This is the fully simplified answer.
 
Fraction CoefficientsAs if square roots weren't bad enough, some teachers like to throw in arithmetic involving fraction exponents, both implied and explicitly written. Just be careful and remember that adding and subtracting like terms is just like counting.
 
Example 5$$\frac{\sqrt{5}}{3} - \sqrt{20}$$$\blacktriangleright$ First let's simplify $\sqrt{20}$:$$\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}$$So now the problem really says$$\frac{\sqrt{5}}{3} - 2\sqrt{5}$$If you have $1/3$ of something and you subtract $2$ then you have $1/3 - 2$ in total. Another way to make this clear is to use common denominators.$$\frac{\sqrt{5}}{3} - 2\sqrt{5} = \frac{\sqrt{5}}{3} - \frac{6\sqrt{5}}{3}$$$$ = -\frac{5\sqrt{5}}{3}$$
 

Put It To The Test

Example 6Simplify.$$\sqrt{150} - \sqrt{24}$$
Show solution
$\blacktriangleright$ Simplify each root expression on its own and see if they can combine together.$$\sqrt{150} = \sqrt{25 \cdot 6} = 5\sqrt{6}$$$$\sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6}$$Therefore,$$\sqrt{150} - \sqrt{24}$$$$= 5\sqrt{6} - 2\sqrt{6} = 3\sqrt{6}$$
 
Example 7Simplify.$$\sqrt{98} - \sqrt{32}$$
Show solution
$\blacktriangleright$ Once again, simplify first.$$\sqrt{98} = \sqrt{49 \cdot 2} = 7 \sqrt{2}$$$$\sqrt{32} = \sqrt{16 \cdot 2} = 4 \sqrt{2}$$Therefore,$$\sqrt{98} - \sqrt{32}$$$$= 7 \sqrt{2} - 4 \sqrt{2} = 3\sqrt{2}$$
 
Example 8Simplify.$$\sqrt{45x} + 6\sqrt{20x}$$
Show solution
$\blacktriangleright$ The $x$ doesn't interact since it isn't a perfect square, but we will get identical root expressions by paring down the integers under the radicals:$$\sqrt{45x} = \sqrt{9 \cdot 5x} = 3\sqrt{5x}$$$$\sqrt{20x} = \sqrt{4 \cdot 5x} = 2\sqrt{5x}$$$$\sqrt{45x} + 6\sqrt{20x}$$$$\longrightarrow 3\sqrt{5x} + 6 \cdot 2 \sqrt{5x}$$$$ = 3\sqrt{5x} + 12 \sqrt{5x} = 15\sqrt{5x}$$
 
Example 9Simplify.$$\sqrt{128a^5 b} + 3\sqrt{2ab^7}$$
Show solution
$\blacktriangleright$ Simplify each.$$\sqrt{128a^5 b} = \sqrt{64a^4 \cdot 2ab} = 8a^2 \sqrt{2ab}$$$$\sqrt{2ab^7} = \sqrt{b^6 \cdot 2ab} = b^3 \sqrt{2ab}$$Therefore,$$\sqrt{128a^5 b} + 3\sqrt{2ab^7}$$$$ = 8a^2 \sqrt{2ab} + 3b^3\sqrt{2ab}$$$$ = \left( 8a^2 + 3b^3 \right) \sqrt{2ab}$$
 
Example 10Simplify.$$\sqrt{30x^2} - \sqrt{48x^3}$$
Show solution
$\blacktriangleright$$$\sqrt{30x^2} = \sqrt{x^2 \cdot 30} = x\sqrt{30}$$$$\sqrt{48x^3} = \sqrt{16x^2 \cdot 3x} = 4x\sqrt{3x}$$These radicals do not have the same expression under the radical symbol, and therefore cannot be combined. The best answer to report is$$\sqrt{30x^2} - \sqrt{48x^3}$$$$=x\sqrt{30} - 4x\sqrt{3x}$$
 
Lesson Takeaways
  • Understand what we mean when we use the words "like terms" for variable radical expressions
  • Know when we can and cannot combine radicals with addition or subtraction
  • Expect to simplify radical expressions in order to determine whether they can be combined

Lesson Metrics

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Headlines - Every lesson is subdivided into mini sections that help you go from "no clue" to "pro, dude". If you keep getting lost skimming the lesson, start from scratch, read through, and you'll be set straight super fast.

Examples - there is no better way to learn than by doing. Some examples are instructional, while others are elective (such examples have their solutions hidden).

Perils and Pitfalls - common mistakes to avoid.

Mister Math Makes it Mean - Here's where I show you how teachers like to pour salt in your exam wounds, when applicable. Certain concepts have common ways in which teachers seek to sink your ship, and I've seen it all!

Put It To The Test - Shows you examples of the most common ways in which the concept is tested. Just knowing the concept is a great first step, but understanding the variation in how a concept can be tested will help you maximize your grades!

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Special Notes

Definitions and Theorems: These are important rules that govern how a particular topic works. Some of the more important ones will be used again in future lessons, implicitly or explicitly.

Pro-Tip: Knowing these will make your life easier.

Remember! - Remember notes need to be in your head at the peril of losing points on tests.

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

Warning! - Something you should be careful about.

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