RADICALS
Copyright by Ingrid Stewart, Ph.D.  Please Send Questions and Comments to ingrid.stewart@csn.edu.

Learning Objectives - This is what you must know after studying the lecture and doing the practice problems!

1.   Use the vocabulary assocatiated with roots and radicals.
2.   Evaluate radicals resulting in integers.
3.   Rewrite exponential expressions as radicals and vice versa.
4.   Evaluate radicals resulting in irrational numbers.
5.   Apply the Product and Quotient Rule of radicals.
6.   Simplify Square Roots.
7.   Rationalize Denominators.

Introducing Roots and Radicals

Finding a root reverses the operation of finding a power.  The radical symbol indicates this process.  There is also some associated vocabulary.

Evaluating Radicals Resulting in Integers

Finding roots of numbers is synonymous to asking us "to evaluate a radical" or "to evaluate a radical expression." Sometimes, we can find the root(s) of a number by hand.  However, most often we must use a calculator to find the root(s).

Example 1:

Evaluate the radical expression .

means to undo . 

is read as the "third root of 64" or the "cube root of 64".  We call the radical; 64 is called the radicand; and 3 is called the index. 

We find that by having memorized that that . Note that is an integer.

Calculator Tip: 

We can find the cube root (index is 3) by activating the calculator function .  This allows us to find roots of any index.  We first type the index and then we press the 2nd button followed by the caret ^ button. 

Example 2:

Evaluate the radical .

means to undo

is read strictly as the "square root of 9."  Please note when the index is 2, it is customarily left off.

We find that by having memorized that .  Note that is an integer!

Calculator Tip: 

We can find the square root (index is 2) in two different ways.

1.   Noticing that the index is 2, we can use the square root button by pressing the 2nd button followed by the the button. 

Warning:  Left parenthesis will open when we activate , specifically we will see .  After we type in the radicand, we MUST type in the right parenthesis, namely , before pressing ENTER.  

2.   We can also find the square root (index is 2) by activating the calculator function .   Then we type the index 2, next we press the 2nd button followed by the caret ^ button. 

A Word about the Principal Root:

Actually, we know that  and .

So why do we say and NOT ?

BY DEFINITION, a radical of EVEN index strictly asks us to find a positive number. This number is also called the principal root.

Therefore, NEVER EQUALS .

is STRICTLY equal to 3.

Example 3:

Evaluate the radical expression .

means undo .

is read as the "fourth root of 81", where 4 is called the index and 81 is called the radicand.

We find that by having memorized that .  Note that is an integer!

Calculator Tip: 

We again activate to find this root.  We have to first type the index , then press the 2nd button followed by the caret ^ button. 

Example 4:

Evaluate the radicals and and .

Since 1(1)(1)(1) ...(1) = 1 no matter what, a radical expression containing a radicand of 1 always has a value of 1.

That is, and and .

Example 5:

Find the following roots: and and .

Since (0)(0)(0) ...(0) = 0 no matter what, a radical expression containing a radicand of 0 always has a value of 0.

That is, and and .

Example 6:

Find using the calculator.

Since we are dealing with a square root, press the 2nd button and then the button. 

We find that .  It is an integer!

Example 7:

Find using the calculator!

We are asked to find the 5th root of 1024.  Translated this means that we are supposed to find the number that produces a result of 1024 when multiplied by itself five times.

We activate to find roots of any index.  Then we type the index and then we press the 2nd button followed by the caret ^ button.

We find .  It is an integer!

Example 8:

Find using the calculator.

We are asked to find the 3rd root of .  Translated this means that we are supposed to find the number that produces a result of when multiplied by itself three times.

We activate to find roots of any index.  Then we type the index and then we press the 2nd button followed by the caret ^ button.

We find that by having memorized that .  Note that is an integer!

Please note that we are dealing with a radical of ODD index, therefore, the outcome is a real number that can be negative!

Rewriting Exponential Expressions as Radicals and Vice Versa

Radical Property:

If is a real number and is a positive rational number with , then

or

Example 9:

Rewrite the following exponential expressions as radicals and simplify:

(1)

(2)

(3)

Remember, 64 = 4(4)(4) !!!

       OR 

(4)

(5)

However, is not yet simplified. We know that which can be simplied to .

Now we can say,

Evaluating Radicals Resulting in Irrational Numbers

If a is a positive real number that is NOT a perfect square, triple, quadruple, etc, then is irrational. 

NOTE:

An irrational number, as opposed to a rational number, is a number that CANNOT be written as a quotient of two integers! These numbers are neither terminating nor repeating decimals.

Most irrational numbers result from finding roots of certain numbers. However, there are also some irrational numbers that occur naturally. For example, the number Pi with its symbol , which is approximately equal to 3.14 and the number , which is approximately equal to 2.72.

For example,144 is a perfect square and = 12 using the calculator, where 12 is a rational number. 

However, 3 is NOT a perfect square and using the calculator, where 1.7320508 ..... is an irrational number.

Example 10:

Evaluate using the calculator.

We press the 2nd button and then the button. 

Warning:  Left parenthesis will open when we activate .  After we type in the radicand, we MUST press the right parenthesis button before we press ENTER button.

We find .  Note that is an IRRATIONAL NUMBER. 

WHY?

The calculator actually does not tell us that is an irrational number.  It fills up all available slots on its screen with decimal places. 

However, from experience we know, that NO whole number multiplied by itself two times (index is 2) gives the result of 24.  Therefore, we know that must be an irrational number.

Example 11:

Find the decimal approximation of the numberusing the calculator.  Round this number to 3 decimal places.

All calculators have a key.  You must know how to find it on your calculator and figure out the key strokes:

Example 12:

Evaluate using the calculator.

First we type the index 3 and then we press the 2nd button followed by the caret ^ button.

We find that .  Note that is an IRRATIONAL NUMBER. 

WHY?

The calculator actually does not tell us that is an irrational number.  It fills up all available slots on its screen with decimal places.

However, from experience we know, that NO whole number multiplied with itself three times (index is 3) gives the result of 25. Therefore, we know that must be an irrational number

Evaluating Radicals Resulting in Imaginary Numbers

Radical Property:

If a is a negative real number, then is an imaginary number, if the index n is even.

Example 13:

Find using the calculator.

Given index 2, we can use the following calculator input:

The calculator tells us Domain Error!

Think about this for a moment!  We are asked to find the number, that when squared, will result in 64.

We know that multiplying two negative numbers results in a positive number and multiplying two positive numbers also results in a positive number.  The only time we get a negative product is when we multiply a negative number with a positive number.

But when we square a number, necessarily, it is either negative or positive.  Therefore, its product is always positive.

Thus, we can come to the conclusion that NO NUMBER EXISTS that, when squared, results in a negative product.

Actually, is called an imaginary number!

Simplify .

Factor 24 into prime factors.

2 is the smallest prime number and it is a factor of 24, that is, 24 = 2 12

2 is also a factor of 12, that is, 12 = 2 6

2 is also a factor of 6, that is, 6 = 2 3

3 is a prime number, that is 3 = 3 1

We are done with the prime factorization process when the multiplier equals 1!

This shows that 24 can be written as a product of prime factors as follows:

We will now write

Next, we will use the Product Rule to separately write the factors of the radicand.

We know that , therefore, we can say that

Please note that .  The multiplication sign between a rational and an irrational number is usually not written.  However, you must understand that the two numbers are connected by MULTIPLICATION.

The number that multiplies a square root is called a coefficient.  For example, in , 2 is the coefficient of the square root.

Example 16:

Simplify .

Factor 108 into prime factors.

2 is the smallest prime number and it is a factor of 108, that is, 108 = 2 54<

2 is a factor of 54, that is, 54 = 2 27

3 is the next prime we can use, that is, 27 =  3 9 

3 is a factor of 9, that is, 9 = 3 3

3 is a prime number, that is, 3 = 3 1

We are done with the prime factorization process when the multiplier equals 1!

This shows that 108 can be written as a product of prime factors as follows:

We will now write

Next, we will use the Product Rule to separately write the factors of the radicand.

We know that and , therefore, we can say that .

Example 17:

Simplify if possible.

We know that can be written as . 

Next, we will use the Product Rule to separately write the factors of the radicand.

We know that , therefore, we can say that . Please note that we always write the irrational factor AFTER the rational factor.

Example 18:

Simplify , if possible. 

We know that can be written as . 

Since both 3 and 5 are prime numbers, we cannot simplify further and find that is already simplified as much as possible.

Example 19:

Simplify , if possible. 

First, we will use the Quotient Rule to separately write the numerator and denominator of the radicand.

We know that and . 

Therefore,

Rationalizing Denominators

In mathematics, a common way of "standardizing" the form of a fraction containing a radical in the denominator is "to remove" the radical from the denominator.  The process of "removing" radicals from a denominator is called "rationalizing the denominator."  This is done by multiplying the fraction by a form of the number 1.

Please understand that writing for example, is NOT INCORRECT, but it is usually not done in algebra. However, in trigonometry radicals are often left in the denominator!

Example 20:

Rationalize the denominator of . 

We will rationalize this denominator by multiplying the fraction by an "appropriate" form of the number 1, namely  .

This can be written as .

Then we will use the Product Rule , to write

Since , we can state .