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Guide allo studio > ALGEBRA / TRIG I

Adding and Subtracting Radicals

Learning Outcomes

  • Identify radicals that can be added or subtracted
  • Add and subtract radical expressions
Adding and subtracting radicals is much like combining like terms with variables.  We can add and subtract expressions with variables like this:

5x+3y−4x+7y=x+10y5x+3y - 4x+7y=x+10y

There are two keys to combining radicals by addition or subtraction: look at the index, and look at the radicand. If these are the same, then addition and subtraction are possible. If not, then you cannot combine the two radicals. Remember the index is the degree of the root and the radicand is the term or expression under the radical. In the diagram below, the index is n, and the radicand is 100100.  The radicand is placed under the root symbol and the index is placed outside the root symbol to the left:
nth root with 100 as the radicand and the word Index and radicand
In the graphic below, the index of the expression 12xy312\sqrt[3]{xy} is 33 and the radicand is xyxy. Screen Shot 2016-07-29 at 4.04.52 PM
Practice identifying radicals that are compatible for addition and subtraction by looking at the index and radicand of the roots in the following example.

Example

Identify the roots that have the same index and radicand. 106 10\sqrt{6} −163 -1\sqrt[3]{6} 25 \sqrt{25} 126 12\sqrt{6} 12253 \frac{1}{2}\sqrt[3]{25} −763 -7\sqrt[3]{6}

Answer: Let's start with 106 10\sqrt{6}.  The index is 22 because no root was specified, and the radicand is 66. The only other radical that has the same index and radicand is 126 12\sqrt{6}. −163 -1\sqrt[3]{6} has an index of 33, and a radicand of 66. The only other radical that has the same index and radicand is −763 -7\sqrt[3]{6}. 25 \sqrt{25} has an index of 22 and a radicand of 2525.  There are no other radicals in the list that have the same index and radicand. 126[/latex]hasthesameindexandradicandas [latex]106 12\sqrt{6}[/latex] has the same index and radicand as [latex]10\sqrt{6} 12253 \frac{1}{2}\sqrt[3]{25} has an index of 33 and a radicand of 2525.  There are no other radicals in the list that share these. −763[/latex]hasthesameindexandradicandas [latex]−163 -7\sqrt[3]{6}[/latex] has the same index and radicand as [latex] -1\sqrt[3]{6}

Making sense of a string of radicals may be difficult. One helpful tip is to think of radicals as variables, and treat them the same way. When you add and subtract variables, you look for like terms, which is the same thing you will do when you add and subtract radicals. Let’s use this concept to add some radicals. In this first example, both radicals have the same radicand and index.

Example

Add. 311+711 3\sqrt{11}+7\sqrt{11}

Answer: The two radicals have the same index and radicand. This means you can combine them as you would combine the terms 3a+7a 3a+7a.

311 + 711 \text{3}\sqrt{11}\text{ + 7}\sqrt{11}

Answer

311+711=1011 3\sqrt{11}+7\sqrt{11}=10\sqrt{11}

It may help to think of radical terms with words when you are adding and subtracting them. The last example could be read "three square roots of eleven plus 77 square roots of eleven".   This next example contains more addends, or terms that are being added together. Notice how you can combine like terms (radicals that have the same root and index) but you cannot combine unlike terms.

Example

Add. 52+3+43+22 5\sqrt{2}+\sqrt{3}+4\sqrt{3}+2\sqrt{2}

Answer: Rearrange terms so that like radicals are next to each other. Then add.

52+22+3+43 5\sqrt{2}+2\sqrt{2}+\sqrt{3}+4\sqrt{3}

Answer

52+3+43+22=72+53 5\sqrt{2}+\sqrt{3}+4\sqrt{3}+2\sqrt{2}=7\sqrt{2}+5\sqrt{3}

Notice that the expression in the previous example is simplified even though it has two terms: 72 7\sqrt{2} and 53 5\sqrt{3}. It would be a mistake to try to combine them further! (Some people make the mistake that 72+53=125 7\sqrt{2}+5\sqrt{3}=12\sqrt{5}. This is incorrect because2 \sqrt{2} and 3\sqrt{3} are not like radicals so they cannot be added.)

Example

Add. 3x+12xy3+x 3\sqrt{x}+12\sqrt[3]{xy}+\sqrt{x}

Answer: Rearrange terms so that like radicals are next to each other. Then add.

3x+x+12xy3 3\sqrt{x}+\sqrt{x}+12\sqrt[3]{xy}

Answer

3x+12xy3+x=4x+12xy3 3\sqrt{x}+12\sqrt[3]{xy}+\sqrt{x}=4\sqrt{x}+12\sqrt[3]{xy}

In the following video, we show more examples of how to identify and add like radicals. https://youtu.be/ihcZhgm3yBg

Try It

[ohm_question]3493[/ohm_question]
Sometimes you may need to add and simplify the radical. If the radicals are different, try simplifying first—you may end up being able to combine the radicals at the end, as shown in these next two examples.

Example

Add and simplify. 2403+1353 2\sqrt[3]{40}+\sqrt[3]{135}

Answer: Simplify each radical by identifying perfect cubes.

28⋅53+27⋅532(2)3⋅53+(3)3⋅532(2)33⋅53+(3)33⋅53 \begin{array}{r}2\sqrt[3]{8\cdot 5}+\sqrt[3]{27\cdot 5}\\2\sqrt[3]{{{(2)}^{3}}\cdot 5}+\sqrt[3]{{{(3)}^{3}}\cdot 5}\\2\sqrt[3]{{{(2)}^{3}}}\cdot \sqrt[3]{5}+\sqrt[3]{{{(3)}^{3}}}\cdot \sqrt[3]{5}\end{array}

Simplify.

2⋅2⋅53+3⋅53 2\cdot 2\cdot \sqrt[3]{5}+3\cdot \sqrt[3]{5}

Add.

453+3534\sqrt[3]{5}+3\sqrt[3]{5}

 

Answer

2403+1353=753 2\sqrt[3]{40}+\sqrt[3]{135}=7\sqrt[3]{5}

Example

Add and simplify. xxy43+yx4y3 x\sqrt[3]{x{{y}^{4}}}+y\sqrt[3]{{{x}^{4}}y}

Answer: Simplify each radical by identifying perfect cubes.

xx⋅y3⋅y3+yx3⋅x⋅y3xy33⋅xy3+yx33⋅xy3xy⋅xy3+xy⋅xy3\begin{array}{r}x\sqrt[3]{x\cdot {{y}^{3}}\cdot y}+y\sqrt[3]{{{x}^{3}}\cdot x\cdot y}\\x\sqrt[3]{{{y}^{3}}}\cdot \sqrt[3]{xy}+y\sqrt[3]{{{x}^{3}}}\cdot \sqrt[3]{xy}\\xy\cdot \sqrt[3]{xy}+xy\cdot \sqrt[3]{xy}\end{array}

Add like radicals.

xyxy3+xyxy3 xy\sqrt[3]{xy}+xy\sqrt[3]{xy}

Answer

xxy43+yx4y3=2xyxy3 x\sqrt[3]{x{{y}^{4}}}+y\sqrt[3]{{{x}^{4}}y}=2xy\sqrt[3]{xy}

The following video shows more examples of adding radicals that require simplification. https://youtu.be/S3fGUeALy7E

Subtracting Radicals

Subtraction of radicals follows the same set of rules and approaches as addition—the radicands and the indices (plural of index) must be the same for two (or more) radicals to be subtracted.  In the examples that follow, subtraction has been rewritten as addition of the opposite.

Example

Subtract. 513−313 5\sqrt{13}-3\sqrt{13}

Answer: The radicands and indices are the same, so these two radicals can be combined.

513−313 5\sqrt{13}-3\sqrt{13}

Answer

513−313=213 5\sqrt{13}-3\sqrt{13}=2\sqrt{13}

Example

Subtract. 45a3−3a3−25a3 4\sqrt[3]{5a}-\sqrt[3]{3a}-2\sqrt[3]{5a}

Answer: Two of the radicals have the same index and radicand, so they can be combined. Rewrite the expression so that like radicals are next to each other.

45a3−3a3−25a345a3−25a3)−3a3) 4\sqrt[3]{5a}-\sqrt[3]{3a}-2\sqrt[3]{5a}\\4\sqrt[3]{5a}-2\sqrt[3]{5a})-\sqrt[3]{3a})

Combine. Although the indices of 25a3 2\sqrt[3]{5a} and −3a3 -\sqrt[3]{3a} are the same, the radicands are not—so they cannot be combined.

25a3−3a3) 2\sqrt[3]{5a}-\sqrt[3]{3a})

Answer

45a3−3a3−25a3=25a3−3a3 4\sqrt[3]{5a}-\sqrt[3]{3a}-2\sqrt[3]{5a}=2\sqrt[3]{5a}-\sqrt[3]{3a}

In the video examples that follow, we show more examples of how to add and subtract radicals that don't need to be simplified beforehand. https://youtu.be/5pVc44dEsTI https://youtu.be/77TR9HsPZ6M

Example

Subtract and simplify. 5a5b4−a16ab4 5\sqrt[4]{{{a}^{5}}b}-a\sqrt[4]{16ab}, where a≥0a\ge 0 and b≥0b\ge 0

Answer: Simplify each radical by identifying and pulling out powers of 44.

5a4⋅a⋅b4−a(2)4⋅a⋅b45⋅aa⋅b4−a⋅2a⋅b45aab4−2aab4\begin{array}{r}5\sqrt[4]{{{a}^{4}}\cdot a\cdot b}-a\sqrt[4]{{{(2)}^{4}}\cdot a\cdot b}\\5\cdot a\sqrt[4]{a\cdot b}-a\cdot 2\sqrt[4]{a\cdot b}\\5a\sqrt[4]{ab}-2a\sqrt[4]{ab}\end{array}

The answer is 3aab43a\sqrt[4]{ab}.

In our last videos, we show more examples of subtracting radicals that require simplifying. https://youtu.be/6MogonN1PRQ   https://youtu.be/tJk6_7lbrlw

Summary

Combining radicals is possible when the index and the radicand of two or more radicals are the same. Radicals with the same index and radicand are known as like radicals. It is often helpful to treat radicals just as you would treat variables: like radicals can be added and subtracted in the same way that like variables can be added and subtracted. Sometimes, you will need to simplify a radical expression before it is possible to add or subtract like terms.

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