Section 6.1 Square and th Root Properties
In this section, we discuss what radical expressions like and mean, and learn some properties that allow these expressions to be simplified.
Subsection 6.1.1 Square Roots
Consider the non-negative number You can ask the question “what number multiplies by itself to make ” We use the symbol to represent the answer to this question, even if you know the “answer” is
A way to visualize this question is with a square that has units of area inside it. What would a side length have to be? We use to represent that side length.
Definition 6.1.2. Square Root.
Given a non-negative number if for some positive number then is called the square root of and we can write instead of The symbol is called the radical or the root. We call expressions with the symbol radical expressions. The number inside the radical is called the radicand.
For example, if you see the expression you should think about the equation (or if you prefer, the equation ) and ask yourself if you know a non-negative value for that solves that equation. Of course, is a non-negative solution. So we can say
To demonstrate vocabulary, both and are radical expressions. In both expressions, the number is the radicand.
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 Just knowing that fact from memory lets us know that It’s advisable to memorize the following:
Subsection 6.1.2 Square Root Decimal Values
Most square roots have decimal expressions that go on forever. Take as an example. The number is between two perfect squares, and Therefore, as demonstrated in Figure 5, In other words,
With a calculator, we can see:
Actually the decimal will not terminate, and we can write with the symbol instead of an equal sign. To get we rounded down slightly from the true value of With a calculator, we can check that a little less than 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 without a calculator, we can find the nearest perfect squares that are whole numbers on either side of Recall that the perfect squares are The perfect square that is just below is and the perfect square just above is
This tells us that is between and or between and We can also say that is much closer to than because is closer to so we think or would be a good estimate.
To check our estimates ( or ) we can square them and see if the result is close to We find and so our estimates are pretty good.
Checkpoint 6.1.7.
Estimate without a calculator.
Explanation.
The radicand, is between and so is between and or between and We notice that is in the middle between and but closer to We estimate to be about
We can check our estimate by calculating:
So is close to and is close to
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 and we would like to know what number could be multiplied with itself three times to make
A way to visualize this question is to imagine a cube with units of volume inside it. What would an edge length have to be? We use to represent that edge length.
Definition 6.1.8. th Root.
- When
is odd, there is always exactly one real number th root for any and we write to mean that one th root. - When
is even and is positive, there are two real number th roots, one of which is positive and the other of which is negative. We write to mean the positive th root. - When
is even and is negative, there aren’t any real number th roots, and we say that is “not a real number”. - When
is then is the th root. On other words,
The symbol is called the th radical or the th root. We call expressions with the symbol radical expressions. The number inside the radical is called the radicand. The index of a radical is the number in
As we noted earlier, when we have we can say “cube root” instead of “ rd root”. Also, when we have we usually say “square root” instead of “ nd root” and we usually just write without the “ ”.
As with square roots, in general an th root’s decimal value is a decimal that goes on forever. For example, For practical applications, we may want to use a calculator to find a decimal approximation to an th root. Some calculators will do this for you directly, and some will not.
- Maybe your calculator has a button that looks like
Then you should be able to type something like3
20
to get - Maybe your calculator has a button that looks like
Then you should be able to type something like3
20
to get - Maybe your calculator allows you to type letters, parentheses, and commas, and you can type
root(3,20)
to get - Maybe your calculator allows you to type letters, parentheses, and commas, but the syntax for an
th root is reversed from the last example, and you can typeroot(20,3)
to get - If your calculator has none of the above options, then you should be able to type
20^(1/3)
as a way to get This is technically using mathematics we will learn later in Section 3.
Try using your own calculator to calculate so that you can become familiar with whatever method it uses.
Fact 6.1.9.
Example 6.1.10.
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 of the pyramid, in in3, is given by where 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?
Subsection 6.1.4 Roots of Negative Numbers
As noted in Definition 8, when the index of an th root is odd, there will always be a real number th root even when the radicand is negative. For example, is because
When the index of an th root is even, it is a problem to have a negative radicand. For example, to find you would need to find a value so that But whether is positive or negative, multiplying by itself will give a positive result. It could never be 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 th root with a negative radicand, such as An even-indexed root of a negative number is not a real number.
If you are confronted with an expression like or (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 and 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.)
Knowing these algebra properties helps to make complicated radical expressions look simpler.
Example 6.1.13.
Checkpoint 6.1.14.
Simplify
Explanation.
As with the previous example, it will help if can be written as a product of a perfect square. In this case, divides and So
But we aren’t done. Can can be written as a product of a perfect square? Yes, because So
This is as simple as we can make this expression.
Example 6.1.15.
Simplify 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 as the product of two numbers in a helpful way. Since we have a cube root, writing as a product of a perfect cube would be helpful. We can write where is a perfect cube.
Checkpoint 6.1.16.
Simplify
Explanation.
As with the previous example, it will help if can be written as a product of a th power. In this case, divides and So
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.
Example 6.1.17.
This is as simple as we can make this expression, unless you prefer to write it as
Checkpoint 6.1.18.
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
Explanation.
We will simplify each radical first, and then multiply them together. We do not want to multiply because we would end up with a larger number that is harder to factor.
It is worth noting that this is considered as simple as we can make it, because the radicand of is so small.
Checkpoint 6.1.20.
Multiply
Explanation.
First multiply the non-radical factors together and the radical factors together. Then look for further simplifications.
Example 6.1.21.
Multiply
Explanation.
First multiply the fractions together under the radical. Then look for further simplifications.
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.
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 combines two like terms, combines two like radicals.
Example 6.1.22.
Simplify
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.
Checkpoint 6.1.23.
Simplify
Explanation.
First we will simplify the radical term where is the radicand, and we may see that we then have like radicals.
Example 6.1.24.
Simplify
Explanation.
We cannot simplify the expression further because and are not like radicals.
Example 6.1.25.
Simplify
Explanation.
In this example, we should multiply the latter two square roots first (after simplifying them) and then see if we have like radicals.
Subsection 6.1.8 Distributing with Square Roots
In Section 5.4, we learned how to multiply polynomials like and 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
Explanation.
We will use the distributive property to do this problem:
Example 6.1.27.
Multiply
Explanation.
We will use the FOIL Method to expand the product. There is an opportunity to simplify some of the radicals after multiplying.
Example 6.1.28.
Expand
Explanation.
We will use the FOIL method to expand this expression:
Example 6.1.29.
Multiply
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 and will simplify to )
Reading Questions 6.1.9 Reading Questions
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2.
Choose one of the radical properties from List 12. Then find its counterpart in the exponent properties from List 5.6.13.
3.
Describe a way you can visualize in a geometric shape. Describe a way you can visualize 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.
Exercise Group.
Evaluate the expression without using a calculator.
Exercise Group.
Without using a calculator...
Simplify Radical Expressions.
Evaluate the following.
Simplifying Square Root Expressions.
Simplify the expression or state that it is not a real number.
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Higher Index Roots.
Simplify the higher index radical.
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