quare Roots: Introduction & Simplification

Roots" (or "radicals") are the "opposite" operation of applying exponents; you can "undo" a power with a radical, and a radical can "undo" a power. For instance, if you square 2, you get 4, and if you "take the square root of 4", you get 2; if you square 3, you get 9, and if you "take the square root of 9", you get 3: Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved



The "" symbol is called the "radical"symbol. (Technically, just the "check mark" part of the symbol is the radical; the line across the top is called the "vinculum".) The expression " " is read as "root nine", "radical nine", or "the square root of nine".
You can raise numbers to powers other than just 2; you can cube things, raise them to the fourth power, raise them to the 100th power, and so forth. In the same way, you can take the cube root of a number, the fourth root, the 100th root, and so forth. To indicate some root other than a square root, you use the same radical sybmol, but you insert a number into the radical, tucking it into the "check mark" part. For instance:



The "3" in the above is the "index" of the radical; the "64" is "the argument of the radical", also called "the radicand". Since most radicals you see are square roots, the index is not included on square roots. While " " would be technically correct, I've never seen it used.

a square (second) root is written as
a cube (third) root is written as
a fourth root is written as
a fifth root is written as:

You can take any counting number, square it, and end up with a nice neat number. But the process doesn't always work going backwards. For instance, consider , the square root of three. There is no nice neat number that squares to 3, so cannot be simplified as a nice whole number. You can deal with in either of two ways: If you are doing a word problem and are trying to find, say, the rate of speed, then you would grab your calculator and find the decimal approximation of :



Then you'd round the above value to an appropriate number of decimal places and use a real-world unit or label, like "1.7 ft/sec". On the other hand, you may be solving a plain old math exercise, something with no "practical" application. Then they would almost certainly want the "exact" value, so you'd give your answer as being simply "".

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Simplifying Square-Root Terms
To simplify a square root, you "take out" anything that is a "perfect square"; that is, you take out front anything that has two copies of the same factor:





Note that the value of the simplified radical is positive. While either of +2 and –2 might have been squared to get 4, "the square root of four" is defined to be only the positive option, +2. When you solve the equation x2 = 4, you are trying to find all possible values that might have been squared to get 4. But when you are just simplifying the expression , the ONLY answer is "2"; this positive result is called the "principal" root. (Other roots, such as –2, can be defined using graduate-school topics like "complex analysis" and "branch functions", but you won't need that for years, if ever.)
Sometimes the argument of a radical is not a perfect square, but it may "contain" a square amongst its factors. To simplify, you need to factor the argument and "take out" anything that is a square; you find anything you've got a pair of inside the radical, and you move it out front. To do this, you use the fact that you can switch between the multiplication of roots and the root of a multiplication. In other words, radicals can be manipulated similarly to powers:



Simplify There are various ways I can approach this simplification. One would be by factoring and then taking two different square roots:



The square root of 144 is 12.

You probably already knew that 122 = 144, so obviously the square root of 144 must be 12. But my steps above show how you can swich back and forth between the different formats (multiplication inside one radical, versus multiplication of two radicals) to help in the simplification process.

Simplify Neither of 24 and 6 is a square, but what happens if I multiply them inside one radical?



Simplify


This answer is pronounced as "five, root three". It is proper form to put the radical at the end of the expression. Not only is "" non-standard, it is very hard to read, especially when hand-written. And write neatly, because "" is not the same as "".
You don't have to factor the radicand all the way down to prime numbers when simplifying. As soon as you see a pair of factors or a perfect square, you've gone far enough.

Simplify Since 72 factors as 2×36, and since 36 is a perfect square, then:



Since there had been only one copy of the factor 2 in the factorization 2×6×6, that left-over 2 couldn't come out of the radical and had to be left behind.

Simplify


More Simplification / Multiplication (page 2 of 7)
Sections: Square roots, More simplification / Multiplication, Adding (and subtracting) square roots, Conjugates / Dividing by square roots, Rationalizing denominators, Higher-Index Roots, A special case of rationalizing / Radicals & exponents / Radicals & domains

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Variables in a radical's argument are simplified in the same way: whatever you've got a pair of can be taken "out front".

Simplify


Simplify The 12 is the product of 3 and 4, so I have a pair of 2's but a 3 left over. Also, I have two pairs of a's; three pairs of b's, with one b left over; and one pair of c's, with one c left over. So the root simplifies as:



You are used to putting the numbers first in an algebraic expression, followed by any variables. But for radical expressions, any variables outside the radical should go in front of the radical, as shown above.

Simplify Writing out the complete factorization would be a bore, so I'll just use what I know about powers. The 20 factors as 4×5, with the 4 being a perfect square. The r18 has nine pairs of r's; the s is unpaired; and the t21 has ten pairs of t's, with one t left over. Then:




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Technical point: Your textbook may tell you to "assume all variables are positive" when you simplify. Why? The square root of the square of a negative number is not the original number. For instance, you could start with –2, square to get +4, and then take the square root (which is defined to be the positive root) to get +2. You plugged in a negative and ended up with a positive. Sound familiar? It should: it's how the absolute value works: |–2| = +2. Taking the square root of the square is in fact the technical definition of the absolute value. But this technicality can cause difficulties if you're working with values of unknown sign; that is, with variables. The |–2| is +2, but what is the sign on | x |? You can't know, because you don't know the sign of x itself — unless they specify that you should "assume all variables are positive", or at least non-negative (which means "positive or zero").

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Multiplying Square Roots
The first thing you'll learn to do with square roots is "simplify" terms that add or multiply roots.
Simplifying multiplied radicals is pretty simple. We use the fact that the product of two radicals is the same as the radical of the product, and vice versa.

Write as the product of two radicals:
Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved

Okay, so that manipulation wasn't very useful. But working in the other direction can be helpful:

Simplify by writing with no more than one radical:


Simplify by writing with no more than one radical:


Simplify by writing with no more than one radical:


The process works the same way when variables are included:

Simplify by writing with no more than one radical:


Adding (and Subtracting) Square Roots (page 3 of 7)
Sections: Square roots, More simplification / Multiplication, Adding (and subtracting) square roots, Conjugates / Dividing by square roots, Rationalizing denominators, Higher-Index Roots, A special case of rationalizing / Radicals & exponents / Radicals & domains

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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 1999-2009 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