RADICALS, ROOTS & POWERS - A Guide of Hints, Strategies and Simple Explanations - Algebra in Words

Algebra in Words: A Guide of Hints, Strategies and Simple Explanations (2014)

RADICALS, ROOTS & POWERS

Note: Before beginning, I must stress (again) that radicals may resemble long division, but they are completely different. Students sometimes mix them up and try to apply long division to radicals, but dealing with radicals is not division. Pay special attention to treat radicals as their own unique function.

Why do you need to understand radicals? One of the main needs and uses is for solving quadratic equations using the Quadratic Formula. They are also commonly applied in problems using the Pythagorean Theorem (which is not covered in this book).

Why is using radicals easy? Because they are nothing more than simple rearrangements of factors. You just have to know what you’re looking for. I will tell you what to look for in the coming pages.

When it comes to dealing with radicals, the main objective is to simplify them. To simplify them, you must be able to toggle between different versions of them, and rearrange them. To do that, you must have a good handle onperfect squares and factoring.

Perfect Squares & Associated Square Roots

You can take the square root of any (positive) number or term, but the result may not be an integer. Perfect squares are numbers whose square roots are integers (non-decimal numbers). A set of very common perfect squares are listed after the following explanation.

The list of radicals about to be listed follow the relationship seen below:

.

In words: When the root of a radical is the same as the power of the base of the radicand, the radical simplifies to that base (which, here, is “x”). Also, if a radical is raised to the same number as the root, it also simplifies to equal the base (again, here, is “x”). The important thing to realize is that both simplify to the same base, “x,” in this case. This equality also exemplifies another property, which is that the exponent can be moved from the radicand to outside the radical, and vice versa. This is a necessary manipulation technique.

Observe this example using 7 as the same root and power:

In words: The seventh root of base x to the power of seven equals x; and:

The seventh root of x in parentheses, raised to the power of seven (outside the parentheses) also equals x.

Notice in both versions, the root is the same as the exponent, and both versions equal “x”. You will notice this trend in the list of perfect squares and square roots, in the next pages.

This also follows in the next example where the root and power are both 2. You may notice there is no “2” written as the root, but don’t let that deceive you. For square roots, the root 2 is usually not written, but it’s also not wrong if you write it in.

In words: The square root of a radicand whose base is squared equals that base.

Equivalently: The square of the square root of some radicand equals that radicand.

Notice: whether the square is inside the radical or outside the radical, the result is the same.

Before working with (simplifying) radicals, it is important to know some of the common perfect squares. In the next list, the left column shows the squares, and to the right shows the associated square roots. Having these memorized will help you simplify radicals more quickly and easily. These are listed as an easy reference, but also so you can see the patterns as discussed above.

List of Perfect Squares & Associated Square Roots

02 = 0;   = 0

12 = 1;   = +/-1

22 = 4;   = +/-2

32 = 9;   = +/-3

42 = 16; = +/-4

52 = 25; = +/-5

62 = 36; = +/-6

72 = 49; = +/-7

82 = 64; = +/-8

92 = 81; = +/-9

102 = 100; = +/-10

112 = 121; = +/-11.

You get the idea…

122 = 144

132 = 169

142 = 196

152 = 225

162 = 256

172 = 289

182 = 324

192 = 361

202 = 400

Common Perfect Cubes & Associated Cube Roots

The following is a list of common perfect cubes and their associated cube roots. They are here for you to reference, but you should also memorize them and notice the patterns of moving the exponent in and out of the radical as discussed a few pages ago.

03 = 0; = 0

13 = 1; = 1

-13 = -1; = -1

23 = 8; = 2

33 = 27; = 3

43 = 64; = 4

53 = 125; = 5

63 = 216

73 = 343

83 = 512

93 = 729

103 = 1000

Other Powers & Relationships of 2, 3, 4 & 5

This is an extra section to show other common powers and exponential relationships for base 2 through base 5, at powers of 4 and 5. Notice here how these numbers can be factored and rearranged using the rules of multiplying bases with exponents and taking powers of powers.

24 = (22)(22) = = (4)(4) = 16

34 = (32)(32) = = (9)(9) = 81

44 = (42)(42) = = (16)(16) = 256

54 = (52)(52) = = (25)(25) = 625

25 = (22)(23) = = = 32

35 = (32)(33) = = = 243

45 = (42)(43) = = = 1024

55 = (52)(53) = = = 3125

These lists are here for your reference. Keep these power and root relationships in mind for the next section, as they play a helpful role in manipulating and simplifying radicals.

Manipulating & Simplifying Radicals

The reason it’s important to be able to recall the perfect squares, perfect cubes, and other perfect powers is because they are essential in simplifying radicals, which is why I listed many of them in the previous section. One major reason for simplifying radicals is to find which radicals are like-terms, so they may be combined as you would combine like-terms with variables (and there are other reasons, too).

It is important to know that simplifying radicals is different than simplifying other terms or expressions that don’t have radicals, so you can’t expect to use the same strategy. The main difference is that when simplifying non-radical terms or expressions, you usually resort to factoring into prime factors and/or finding a GCF to factor out. Simplifying radicals actually means factoring and reorganizing factors of the radicand, but the radicand is not necessarily prime factored.

To simplify radicals, you must factor the radicand into two types of factors:

·   perfect-power-factors and

·   non-perfect-power factors.

And there is a very logical reason for this. The radical of the perfect-power is to be taken, and then (its root) will be moved outside the radical (and treated like a coefficient that is multiplied by the remaining radical). The non-perfect-power factor will simply remain under the radical because that is its most simplified form.

Observe this method in the next section Common Radical Fingerprints. Look at the example for the square root of 12. Notice that its factor “4” is a perfect square factor, and “3” is not, so you separate them into factors (4)(3). Now, since 4 is a perfect square, take the square root of it. Since the square root of 4 is 2, the 2 gets moved to the outside of the radical as a coefficient, and the square root of 3 remains in the radical, leaving you with (as you would say) “two (times) the square root of three.”

The reason radicals are simplified this way is so they can be manipulated into like-terms that can be combined (as in “combine like-terms”).

For radicals, “like-terms” are terms in which both the root and radicand are exactly the same. When these criteria are met, like-radicals are combined via their coefficients the same way as like-terms with variables.

As in the last section in which I show a list of common roots and powers, there are others that are still common, but “not perfect”… “not perfect” in the sense that the radical cannot be reduced to an integer. These are so common, that I call them “fingerprints,” because after encountering them enough, you may memorize them, saving you the step of having to manually factor and simplify them every time.

The following list accomplishes three purposes.

1.  It simply shows common “non-perfect” radicals, and

2. it shows the intermediate steps where the radicands are factored into “perfect-powers” (in this case, they’re perfect squares), and “non-perfect powers” (which in this case are non-perfect squares).

3. It also demonstrates the “Product Rule of Radicals.”

I also decided to include a few common radicals which are already in their most reduced form, just to put them into perspective.

List of Common Radical Fingerprints






It is important to note that when doing square roots (or any even roots) on a calculator, most calculators will only report the positive root, so it is up to you to also write the negative.

Extraneous Roots in Radical Equations

The topic of extraneous roots (a.k.a. extraneous solutions) has been explained previously, as well as how to identify them when they are in a denominator. But they can also be in an equation with radicals. When they are, you must check your answer by substituting back into the original equation and simplifying. This is mentioned in an upcoming section: Checking Your Answers.