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Section 6.1 Square and nth Root Properties

In this section, we discuss what radical expressions like 50 and 273 mean, and learn some properties that allow these expressions to be simplified.
Figure 6.1.1. Alternative Video Lesson

Subsection 6.1.1 Square Roots

Consider the non-negative number 25. You can ask the question “what number multiplies by itself to make 25?” We use the symbol 25 to represent the answer to this question, even if you know the “answer” is 5.
A way to visualize this question is with a square that has 25 units of area inside it. What would a side length have to be? We use 25 to represent that side length.

Definition 6.1.2. Square Root.

Given a non-negative number x, if rr=x for some positive number r, then r is called the square root of x, and we can write x instead of r. The x symbol is called the radical or the root. We call expressions with the x symbol radical expressions. The number inside the radical is called the radicand.
For example, if you see the expression 16, you should think about the equation rr=16 (or if you prefer, the equation r2=16) and ask yourself if you know a non-negative value for r that solves that equation. Of course, 4 is a non-negative solution. So we can say 16=4.
To demonstrate vocabulary, both 2 and 32 are radical expressions. In both expressions, the number 2 is the radicand.
The word “radical” means something like “on the fringes” when used in politics, sports, and other places. It actually has that same meaning in math, when you consider a square with area A as in Figure 3.
A drawing of a square labeled with area A; the side lengths are labeled the square root of A
Figure 6.1.3. “Radical” means “off to the side.”
The one-digit multiplication times table has special numbers along the diagonal. They are known as perfect squares. And for working with square roots, it will be helpful if you can memorize these first few perfect square numbers. For example, the times table tells us that 77=49. Just knowing that fact from memory lets us know that 49=7. It’s advisable to memorize the following:
0=0 1=1 4=2
9=3 16=4 25=5
36=6 49=7 64=8
81=9 100=10 121=11
144=12 225=15 256=16
× 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9
2 2 4 6 8 10 12 14 16 18
3 3 6 9 12 15 18 21 24 27
4 4 8 12 16 20 24 28 32 36
5 5 10 15 20 25 30 35 40 45
6 6 12 18 24 30 36 42 48 54
7 7 14 21 28 35 42 49 56 63
8 8 16 24 32 40 48 56 64 72
9 9 18 27 36 45 54 63 72 81
Figure 6.1.4. Multiplication table with squares

Subsection 6.1.2 Square Root Decimal Values

Most square roots have decimal expressions that go on forever. Take 5 as an example. The number 5 is between two perfect squares, 4 and 9. Therefore, as demonstrated in Figure 5, 4<5<9. In other words,
2<5<3
So 5 has a decimal value somewhere between 2 and 3.
Figure 6.1.5. 2<5<3
With a calculator, we can see:
5=2.236
Actually the decimal will not terminate, and we can write 52.236 with the symbol instead of an equal sign. To get 2.236 we rounded down slightly from the true value of 5. With a calculator, we can check that 2.2362=4.999696, a little less than 5, but close.
When the radicand is a perfect square, its square root is a rational number. If the radicand is not a perfect square, the square root is irrational. (It has a decimal that goes on forever without any repeating pattern.) We want to be able to estimate square roots without using a calculator.

Example 6.1.6.

To estimate 10 without a calculator, we can find the nearest perfect squares that are whole numbers on either side of 10. Recall that the perfect squares are 1,4,9,16,25,36,49,64, The perfect square that is just below 10 is 9 and the perfect square just above 10 is 16.
This tells us that 10 is between 9 and 16, or between 3 and 4. We can also say that 10 is much closer to 3 than 4 because 10 is closer to 9, so we think 3.1 or 3.2 would be a good estimate.
To check our estimates (3.1 or 3.2) we can square them and see if the result is close to 10. We find 3.12=9.61 and 3.22=10.24, so our estimates are pretty good.

Checkpoint 6.1.7.

Estimate 19 without a calculator.
Explanation.
The radicand, 19, is between 16 and 25, so 19 is between 16 and 25, or between 4 and 5. We notice that 19 is in the middle between 16 and 25 but closer to 16. We estimate 19 to be about 4.4.
We can check our estimate by calculating:
4.42=19.36
So 4.42 is close to 19, and 4.4 is close to 19.

Subsection 6.1.3 Cube Root and Higher Order Roots

The concept of a square root extends to a cube root. What if we have a number in mind, like 64, and we would like to know what number could be multiplied with itself three times to make 64?
A way to visualize this question is to imagine a cube with 64 units of volume inside it. What would an edge length have to be? We use 643 to represent that edge length.

Definition 6.1.8. nth Root.

Given a number x, if rrrn instances=x for some number r (or if you prefer, rn=x) then r is called an nth root of x.
  • When n is odd, there is always exactly one real number nth root for any x, and we write xn to mean that one nth root.
  • When n is even and x is positive, there are two real number nth roots, one of which is positive and the other of which is negative. We write xn to mean the positive nth root.
  • When n is even and x is negative, there aren’t any real number nth roots, and we say that xn is “not a real number”.
  • When x is 0, then 0 is the nth root. On other words, 0n=0.
The xn symbol is called the nth radical or the nth root. We call expressions with the xn symbol radical expressions. The number inside the radical is called the radicand. The index of a radical is the number n in xn.
As we noted earlier, when we have x3, we can say “cube root” instead of “3rd root”. Also, when we have x2, we usually say “square root” instead of “2nd root” and we usually just write x without the “2”.
Here are some examples of nth roots:
  • 83=2, because 2223 instances=8.
  • 814=3, because 33334 instances=81.
  • 325=2, because (2)(2)(2)(2)(2)5 instances=32.
As with square roots, in general an nth root’s decimal value is a decimal that goes on forever. For example, 203=2.714. For practical applications, we may want to use a calculator to find a decimal approximation to an nth root. Some calculators will do this for you directly, and some will not.
  • Maybe your calculator has a button that looks like yx. Then you should be able to type something like 3 yx 20 to get 2032.714.
  • Maybe your calculator has a button that looks like yn. Then you should be able to type something like 3 yn 20 to get 2032.714.
  • Maybe your calculator allows you to type letters, parentheses, and commas, and you can type root(3,20) to get 2032.714.
  • Maybe your calculator allows you to type letters, parentheses, and commas, but the syntax for an nth root is reversed from the last example, and you can type root(20,3) to get 2032.714.
  • If your calculator has none of the above options, then you should be able to type 20^(1/3) as a way to get 2032.714. This is technically using mathematics we will learn later in Section 3.
Try using your own calculator to calculate 203 so that you can become familiar with whatever method it uses.

Example 6.1.10.

A solution to x5=48 is 485.

Example 6.1.11.

A pyramid has a square base, and its height is equal to one side length of the square at its base. In this situation, the volume V of the pyramid, in in3, is given by V=13s3, where s is the pyramid’s base side length in in. Archimedes dropped the pyramid in a bathtub, and judging by how high the water level rose, the volume of the pyramid is 243 in3 (a little more than 1 gal). How tall is the pyramid?
The equation tells us that:
243=13s3
We can multiply on both sides by 3 and:
729=s3
So we are looking for a solution to the equation s3=729. This means that s is 7293. A calculator tells us that this equals 9. So the pyramid’s height is 9 inches.

Subsection 6.1.4 Roots of Negative Numbers

Can we find the square root of a negative number, like 64? How about its cube root, 643?
As noted in Definition 8, when the index of an nth root is odd, there will always be a real number nth root even when the radicand is negative. For example, 643 is 4, because (4)(4)(4)=64.
When the index of an nth root is even, it is a problem to have a negative radicand. For example, to find 64, you would need to find a value r so that rr=64. But whether r is positive or negative, multiplying r by itself will give a positive result. It could never be 64. So there is no way to have a real number square root of a negative number. And the same thing is true for any even index nth root with a negative radicand, such as 644. An even-indexed root of a negative number is not a real number.
If you are confronted with an expression like 25 or 164 (any square root or even-indexed root of a negative number), you can state that the expression “is not real” or that it is “not defined” (as a real number). Don’t get carried away though. Expressions like 273 and 15 are defined, because the index is odd.

Subsection 6.1.5 Radical Properties and Exponent Properties

In Chapter 5, we learned some algebra properties of exponents. A summary of these properties is in List 5.6.13. There are some similar properties for radicals, presented here without explanation. (These properties are easier to explain once we are in Section 3.)
List 6.1.12. Properties of Radicals for Multiplication and Division
If a and b are positive real numbers, and m is a positive [cross-reference to target(s) "item-integer-definition" missing or not unique], then we have the following properties:
Root of a Product Property
abm=ambm
Root of a Quotient Property
abm=ambm as long as b0
Knowing these algebra properties helps to make complicated radical expressions look simpler.

Example 6.1.13.

Simplify 18. Anything we can do to make the radicand a smaller simpler number is helpful. Note that 18=92, so we can write
18=92=92according to the Root of a Product Property=32
This expression 32 is considered “simpler” than 18 because the radicand is so much smaller.

Checkpoint 6.1.14.

Simplify 72.
Explanation.
As with the previous example, it will help if 72 can be written as a product of a perfect square. In this case, 4 divides 72, and 72=418. So
72=418=418=218
But we aren’t done. Can 18 can be written as a product of a perfect square? Yes, because 18=92. So
72=218=292=292=232=62
This is as simple as we can make this expression.

Example 6.1.15.

Simplify 803. Anything we can do to make the radicand a smaller simpler number is helpful. With lessons learned from the previous examples, maybe there is a way to rewrite 80 as the product of two numbers in a helpful way. Since we have a cube root, writing 80 as a product of a perfect cube would be helpful. We can write 80=810, where 8 is a perfect cube.
803=8103=83103according to the Root of a Product Property=2103
This expression 2103 is considered “simpler” than 803 because the radicand is so much smaller.

Checkpoint 6.1.16.

Simplify 484.
Explanation.
As with the previous example, it will help if 48 can be written as a product of a 4th power. In this case, 16 divides 48, and 48=163. So
484=1634=16434=234
This is as simple as we can make this expression.
When a radical is applied to a fraction, the Root of a Quotient Property is useful.

Checkpoint 6.1.18.

Simplify each of the following.
(b)
8273
Explanation.
8273=83273=23
(c)
811253
Explanation.
811253=8131253=8135=27335=273335=3335

Subsection 6.1.6 Multiplying Square Root Expressions

We can use the Root of a Product Property and the Root of a Quotient Property to multiply and divide square root expressions. We want to simplify each radical first to keep the radicands as small as possible.

Example 6.1.19.

Multiply 854.
Explanation.
We will simplify each radical first, and then multiply them together. We do not want to multiply 854 because we would end up with a larger number that is harder to factor.
854=4296=4296=2236=626=612=643=643=623=123
It is worth noting that this is considered as simple as we can make it, because the radicand of 3 is so small.

Checkpoint 6.1.20.

Multiply 27321.
Explanation.
First multiply the non-radical factors together and the radical factors together. Then look for further simplifications.
27321=23721=6773=6493=673=423

Example 6.1.21.

Multiply 6535.
Explanation.
First multiply the fractions together under the radical. Then look for further simplifications.
6535=6535=1825=1825=925=325

Subsection 6.1.7 Adding and Subtracting Square Root Expressions

We learned the Root of a Product Property previously and applied this to multiplication of square roots, but we cannot apply this property to the operations of addition or subtraction. Here are two examples to demonstrate why not.
9+16=?9+1616925=?169253+4=?25135=?1447=no58=no12
We do not get the same result if we combine radical sums and differences in the same way we can combine radical products and quotiens.
To add and subtract radical expressions, we need to recognize that we can only add and subtract like terms. In this case, we will call them like radicals. Adding like radicals will work just like adding like terms. In the same way that x+3x=4x combines two like terms, 5+35=45 combines two like radicals.

Example 6.1.22.

Simplify 2+8.
Explanation.
First, simplify each radical. Simplifying is the best way to understand whether or not we even have two like radicals that could be combined.
2+8=2+42=2+22=32

Checkpoint 6.1.23.

Simplify 23348.
Explanation.
First we will simplify the radical term where 48 is the radicand, and we may see that we then have like radicals.
23348=233163=23343=23123=103

Example 6.1.24.

Simplify 2+27.
Explanation.
2+27=2+93=2+33
We cannot simplify the expression further because 2 and 3 are not like radicals.

Example 6.1.25.

Simplify 61812.
Explanation.
In this example, we should multiply the latter two square roots first (after simplifying them) and then see if we have like radicals.
61812=69243=63223=6623=666=56

Subsection 6.1.8 Distributing with Square Roots

In Section 5.4, we learned how to multiply polynomials like 2(x+3) and (x+2)(x+3). All the methods we learned there apply when we multiply square root expressions. We will look at a few examples done with different methods.

Example 6.1.26.

Multiply 5(32).
Explanation.
We will use the distributive property to do this problem:
5(32)=5352=1510

Example 6.1.27.

Multiply (6+12)(32).
Explanation.
We will use the FOIL Method to expand the product. There is an opportunity to simplify some of the radicals after multiplying.
(6+12)(32)=6362+123122=1812+3624=3223+626

Example 6.1.28.

Expand (32)2.
Explanation.
We will use the FOIL method to expand this expression:
(32)2=(32)(32)=(3)23223+(2)2=366+2=526

Example 6.1.29.

Multiply (57)(5+7).
Explanation.
We can once again use the FOIL method to expand this expression. (But it is worth noting that this expression is in the special form (ab)(a+b) and will simplify to a2b2.)
(57)(5+7)=(5)2+5775(7)2=5+35357=2

Reading Questions 6.1.9 Reading Questions

1.

Is there a difference between 23 and 32? Explain.

3.

Describe a way you can visualize 81 in a geometric shape. Describe a way you can visualize 273 in a geometric shape.

Exercises 6.1.10 Exercises

Exercise Group.

Which of the following are square numbers? There may be more than one correct answer.
2.
  • 85
  • 36
  • 119
  • 96
  • 121
  • 100

Exercise Group.

Evaluate the expression without using a calculator.

Exercise Group.

Without using a calculator...
13.
Estimate the value of 99:
  • 10.95
  • 10.05
  • 9.95
  • 9.05
14.
Estimate the value of 19:
  • 4.64
  • 3.36
  • 3.64
  • 4.36

Simplify Radical Expressions.

Evaluate the following.

Simplifying Square Root Expressions.

Simplify the expression or state that it is not a real number.
59.
988125125
60.
9818180125
80.
(45+611)(45611)

Higher Index Roots.

Simplify the higher index radical.

Exercise Group.

Find a solution to the equation. Use a radical expression. If there are no solutions, say so.
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