Adding and Subtracting radical expressions

Adding (and Subtracting) Square Roots
Just as with "regular" numbers, square roots can be added together. But you might not be able to simplify the addition all the way down to one number. Just as "you can't add apples and oranges", so also you cannot combine "unlike" radicals. To add radical terms together, they have to have the same radical part.

• Simplify: 
Since the radical is the same in each term (namely, the square root of three), I can combine the terms. I have two copies of the radical, added to another three copies. This gives me five copies:

That middle step, with the parentheses, shows the reasoning that justifies the final answer. You probably won't ever need to "show" this step, but it's what should be going through your mind.

• Simplify: 
The radical part is the same in each term, so I can do this addition. To help me keep track that the first term means "one copy of the square root of three", I'll insert the "understood" "1":

Don't assume that expressions with unlike radicals cannot be simplified. It is possible that, after simplifying the radicals, the expression can indeed be simplified.

• Simplify: 
To simplify a radical addition, I must first see if I can simplify each radical term. In this particular case, the square roots simplify "completely" (that is, down to whole numbers):

• Simplify:  

I have three copies of the radical, plus another two copies, giving me— Wait a minute! I can simplify those radicals right down to whole numbers:

Don't worry if you don't see a simplification right away. If I hadn't noticed until the end that the radical simplified, my steps would have been different, but my final answer would have been the same:

• Simplify:  
I can only combine the "like" radicals, so I'll end up with two terms in my answer:

There is not, to my knowledge, any preferred ordering of terms in this sort of expression, so the expression  should also be an acceptable answer.

• Simplify:     Copyright © Elizabeth Stapel 1999-2011 All Rights Reserved
I can simplify the radical in the first term, and this will create "like" terms:

• Simplify:  
I can simplify most of the radicals, and this will allow for at least a little simplification:

• Simplify:  
These two terms have "unlike" radical parts, and I can't take anything out of either radical. Then I can't simplify the expression any further and my answer has to be:


(expression is already fully simplified)

• Expand:  
To expand (that is, to multiply out and simplify) this expression, I first need to take the square root of two through the parentheses:

As you can see, the simplification involved turning a product of radicals into one radical containing the value of the product (being 2×3 = 6). You should expect to need to manipulate radical products in both "directions".

• Expand:  

• Expand:  
It will probably be simpler to do this multiplication "vertically".

Simplifying gives me:  

By doing the multiplication vertically, I could better keep track of my steps. You should use whatever multiplication method works best for you.

• Simplify

I do the multiplication:

Then I complete the calculations by simplifying:

• Simplify:  

I do the multiplication:

Then I simplify:

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Note in the last example above how I ended up with all whole numbers. (Okay, technically they're integers, but the point is that the terms do not include any radicals.) I multiplied two radical "binomials" together and got an answer that contained no radicals. You may also have noticed that the two "binomials" were the same except for the sign in the middle: one had a "plus" and the other had a "minus". This pair of factors, with the second factor differing only in the one sign in the middle, is very important; in fact, this "same except for the sign in the middle" second factor has its own name:

Given the radical expression , the "conjugate" is the expression .

The conjugate (KAHN-juh-ghitt) has the same numbers but the opposite sign in the middle. So not only is the conjugate of , but is the conjugate of .
When you multiply conjugates, you are doing something similar to what happens with a difference of squares:

a² – b² = (a + b)(a – b)   Copyright © Elizabeth Stapel 1999-2011 All Rights Reserved

When you multiply the factors a + b and a – b, the middle "ab" terms cancel out:

The same thing happens when you multiply conjugates:

We will see shortly why this matters. To get to that point, let's take a look at fractions containing radicals in their denominators.

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Dividing by Square Roots
Just as you can swap between the multiplication of radicals and a radical containing a multiplication, so also you can swap between the division of roots and one root containing a division.

• Simplify:  
I can simplify this by working inside, and then taking the square root:

...or else by splitting the division into two radicals, simplifying, and cancelling:

• Simplify:  

• Simplify:  

This looks very similar to the previous exercise, but this is the "wrong" answer. Why? Because the denominator contains a radical. The denominator must contain no radicals, or else it's "wrong". (Why "wrong" in quotes? Because this issue may matter to your instructor right now, but it probably won't later on. It's like when you were in elementary school and improper fractions were "wrong" and you had to convert everything to mixed numbers instead. But now that you're in algebra, improper fractions are fine, even preferred. Once you get to calculus or beyond, they won't be so uptight about where the radicals are.)

To get the "right" answer, I must "rationalize" the denominator. That is, I must find some way to convert the fraction into a form where the denominator has only "rational" (fractional or whole number) values. But what can I do with that radical-three? I can't take the 3 out, because I don't have a pair of threes.

Thinking back to those elementary-school fractions, you couldn't add them unless they had the same denominators. To create these "common" denominators, you would multiply, top and bottom, by whatever the denominator needed. Anything divided by itself is just 1, and multiplying by 1 doesn't change the value of whatever you're multiplying by the 1. But multiplying that "whatever" by a strategic form of 1 could make the necessary computations possible, such as:

We can use the same technique to rationalize radical denominators.

I could take a 3 out of the denominator if I had two factors of 3 inside the radical. I can create this pair of 3's by multiplying by another copy of root-three. If I multiply top and bottom by root-three, then I will have multiplied the fraction by a strategic form of 1. I won't have changed the value, but simplification will now be possible:

This last form, "five, root-three, divided by three", is the "right" answer they're looking for.

• Simplify:  

Don't stop once you've rationalized the denominator. As the above demonstrates, you should always check to see if something remains to be simplified.

• Simplify:  
This expression is in the "wrong" form, due to the radical in the denominator. But if I try to multiply through by root-two, I won't get anything useful:

Multiplying through by another copy of the whole denominator won't help, either:

But look what happens when I multiply by the same numbers, but with the opposite sign in the middle:   Copyright © Elizabeth Stapel 1999-2011 All Rights Reserved

This multiplication made the radical terms cancel out, which is exactly what I want. This "same numbers but the opposite sign in the middle" thing is the "conjugate" of the original expression. By using the conjugate, I can do the necessary rationalization.

Do not try to reach inside the numerator and rip out the 6 for "cancellation". The only thing that factors out of the numerator is a 3, but that won't cancel with the 2 in the denominator. Nothing cancels!

• Simplify:  
I'll multiply by the conjugate in order to "simplify" this expression. The denominator's multiplication results in a whole number (okay, a negative, but the point is that there aren't any radicals):

The numerator's multiplication looks like this:

Then the simplified (rationalized) form is:

It can be helpful to do the multiplications separately, as shown above. Don't try to do too much at once, and make sure to check for any simplifications when you're done with the rationalization.
Operations with cube roots, fourth roots, and other higher-index roots work similarly to square roots.
Simplifying Higher-Index Terms

• Simplify
Just as I can pull from a square (or second) root anything that I have two copies of, so also I can pull from a fourth root anything I've got four of:

If you have a cube root, you can take out any factor that occurs in threes; in a fourth root, take out any factor that occurs in fours; in a fifth root, take out any factor that occurs in fives; etc.

• Simplify the cube root: 

• Simplify the cube root: 

• Simplify:     Copyright © Elizabeth Stapel 1999-2011 All Rights Reserved

• Simplify:  

• Simplify:  

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Multiplying Higher-Index Roots

• Simplify:  

• Simplify:  

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Adding Higher-Index Roots

• Simplify:  

• Simplify:  

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Dividing Higher-Index Roots

• Simplify:  

• Simplify:  
I can't simplify this expression properly, because I can't simplify the radical in the denominator down to whole numbers:

To rationalize a denominator containing a square root, I needed two copies of whatever factors were inside the radical. For a cube root, I'll need three copies. So that's what I'll multiply onto this fraction:

• Simplify:  
Since 72 = 8 × 9 = 2 × 2 × 2 × 3 ×3, I won't have enough of any of the denominator's factors to get rid of the radical. To simplify a fourth root, I would need four copies of each factor. For this denominator's radical, I'll need two more 3s and one more 2:

A Special Case of Rationalizing
If your class has covered the formulas for factoring the sums and differences of cubes, then you might encounter a special case of rationalizing denominators. The reasoning and methodology are similar to the "difference of squares" conjugate process for square roots.

• Simplify:  
I would like to get rid of the cube root, but multiplying by the conjugate won't help much:

But I can "create" a sum of cubes, just as using the conjugate allowed me to create a difference of squares earlier. Using the fact that a³ + b³ = (a + b)(a² – ab + b²), and letting a = 1 and b equal the cube root of 4, I get:

If I multiply, top and bottom, by the second factor in the sum-of-cubes formula, then the denominator will simplify with no radicals:

Naturally, if the sign in the middle of the original denominator had been a "minus", I'd have applied the "difference of cubes" formula to do the rationalization. This sort of "rationalize the denominator" exercise almost never comes up. But if you see this in your homework, expect one of these on your next test.

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Radicals Expressed With Exponents
Radicals can be expressed as fractional exponents. Whatever is the index of the radical becomes the denominator of the fractional power. For instance:

The second root became a one-half power. A cube root would be a one-third power, a fourth root would be a one-fourth power, and so forth.This conversion process will matter a lot more once you get to calculus. For now, it allows you to simplify some expressions that you might otherwise not have been able to.   Copyright © Elizabeth Stapel 1999-2011 All Rights Reserved

• Express    as a single radical term.
I will convert the radicals to exponential expressions, and then apply exponent rules to combine the factors:

• Simplify:  

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A Few Other Considerations
Usually, we cannot have a negative inside a square root. (The exception is for "imaginary" numbers. If you haven't done the number "i" yet, then you haven't done imaginaries.) So, for instance, is not possible. Do not try to say something like " ", because it's not true: . You must have a positive inside the square root. This can be important for defining and graphing functions.

• Find the domain of the following:
The fact that I have the expression x – 2 inside a square root requires that x – 2 be zero or greater, so I must have x – 2 > 0. Solving, I get:

domain:  x > 2

On the other hand, you CAN have a negative inside a cube root (or any other odd root). For instance:

...because (–2)³ = –8.

• Find the domain of the following:
For , there is NO RESTRICTION on the value of x, because x – 2 is welcome to be negative inside a cube root. Then the domain is:

domain:  all x