MATH 10 Module 4 (Rational Exponents and Radical Expressions)

Angelicum College Quezon City

Mathematics 10

MODULE 4

Rational Exponents

and Radical

Expressions

Prepared by: Reviewed by:

Ms. Roxan S. Villanueva Mr Florben G. Mendoza

Ms. Frances Maureen B. Viado

Endorsed by: Approved by:

Mrs. Maria Urduja C. Galang Dr. Rossani Del Mundo


2

LESSON 1

Rational Exponents

Hello learners! We are now on the first lesson of our fourth module which is

about rational exponents. These are exponents which are rational numbers. A rational

number is any number that can be written in the form a

b, where a and b are integers

and b 0. In other words, rational exponents are fractional exponents.

In this lesson, you are going to simplify expressions having rational exponents.

Similar to the previous lesson, you will still apply the laws of exponents. In addition, your

skills in solving fractions will also be applied. Solving fractions is not that easy as we

know, thus patience and determination are needed.

In this lesson, you will become more patient and determined in every challenge

that will come your way. The way you look and handle your real life problems can be

related to solving rational exponents because it will tell you who you are and what you

are. If you easily give up on solving rational exponents, that will reflect also on how you

treat your real life problems.

After reading and studying this lesson, you will be able to simplify expressions

with rational exponents. Lets start with a pretest to check your prior knowledge about

this lesson. Please answer the pretest below then compare your answers with the

answer key provided by your facilitator..

PRETEST

DIRECTIONS: Copy the questions and simplify each of the following

expressions.

1. 4.

2. 5.

3.


3

In this lesson, we will simplify expressions with rational exponents. In other

words, these are fractional exponents. We will also apply the laws of exponents in

simplifying expressions with rational exponents.

Let us now have some examples that show how to simplify expressions with

rational exponents. Read and analyze each example.

Example 1: Simplify the following expressions.

a.

b.

What did you notice in the examples above? Yes! The Product Rule was applied

in simplifying the expressions. Just add the exponents and then simplify it.


c.

d.

What law of exponents was used in the above examples? Thats right! Power

Rule was again applied to simplify the expressions above. Just multiply the exponents

then simplify them. In example c, the law on Negative Exponents was also applied.

Always remember that in simplifying expressions having rational exponents, the

skills in solving fractions are very important.

Lets have more examples for further understanding. Take a look at the following

examples and analyze them carefully.


4


e.

f.

In the examples above, the exponents were subtracted and simplified.

What law was applied in the examples above? Correct! Quotient Rule was

applied in simplifying the above expressions.


g.

h.

What did you notice in the examples above? The Power Rule for Products and

Quotient was applied in the above examples.

It is really important to be familiar with all the laws of exponents and when to use

them. In this way simplifying expressions with rational exponents will be easy and

convenient. In addition, the skills in solving fractions are also a big help. Lets find out if

you really understood this lesson. Please answer the activity that follows then get the

answer key from your facilitator to check your work.


5

ACITIVITY

DIRECTIONS: Simplify each of the following expressions.

1.

2.

3.

4.

5.

What have you learned so far?

After a thorough discussion of the lesson on simplifying expressions with

rational exponents, let us now summarize what you have learned.

Expressions with rational exponents mean having fractional exponents.

Laws of exponents are needed to simplify expressions with rational

exponents, as well as the skills in solving fractions.

Patience and determination are developed in this lesson, which are

needed in solving real life problems.

Youre now ready to answer the post test on the next page. After which, compare your

answers with the answer key provided by your facilitator. Do your best and have fun answering the post test!


6

POSTTEST


1.

2.

3.

4.

5.


7

LESSON 2

Roots of Real Numbers

Welcome learners! We are now about to learn about radicals. In connection with

the previous lesson, Exponents, we will continue our discussion on rational exponents.

Radicals are used to define rational exponents. Any expression with rational exponents

represents a radical expression.

The expression , which indicates the nth root of , is called a radical. The

symbol is called a radical sign. It is used for the square root of a number. The

number n is called the index of the radical sign, which gives the order of the radical.

The denominator of the rational exponent corresponds to the index of the radical. The

number within the , is called the radicand. This corresponds to the base raised to the

power indicated by the numerator of the exponent.

The roots of a real number can be positive or negative. In addition, to find the

roots of a real number means to find its square root, cube root, fourth root, and so on

depending on the index, or order of the radical. Thus, finding the roots of a number will

be simply by its reverse operations, that is raising a number to a power.

But how is it important to find the roots of real numbers? How can you relate it

into your daily endeavors? Roots of real numbers, especially square roots, are used in

investigating car accidents by the police and creating or designing a robot requires

algebraic manipulations.

Lets see how far you know about this lesson. Please take the pretest below and

check with the answer key provided by your facilitator..

radicand

exponent index


8

PRETEST


1. 6.

2. 7.

3. 8.

4. 9.

5. 10.

Take a look at the figure below. What figure can you see? Correct! It is a square.

It is a 4-sided polygon with four right angles and four equal sides. To obtain the area of

a square, you need to square its side or multiply the length of its side by itself.

In the figure below, the length of the side of a square is 8 cm and its area is 64

cm2. As you notice, 64 is the square of 8 and since the square of 8 is 64, we could say

that 8 is the square root of 64.

Thus, the inverse of squaring is finding a square root.

Area = 64 cm2

8 cm


9

Lets have another figure. What figure can you see? Yes! It is a solid figure

whose sides are all equal. It is called cube. By multiplying the length of the side three

times to itself or by getting the cube of the length of the side, V = s3, we obtain the

volume of the cube.

As you noticed 27 is the cube of 3. Since the cube of 3 is 27, we could say that

the cube root of 27 is 3.

Therefore, finding the cube of a number and getting its cube root are inverse

processes.

In the previous lesson, we learned to solve expressions with rational exponents.

In this module, radicals are used to define rational exponents. Any expression with

rational exponents represents a radical expression.

The roots of a real number can be positive or negative. In addition, to find the

roots of a real number means to find its square root, cube root, fourth root, and so on

depending on the index, or order of the radical. Thus, finding the roots of a number will

be simply by its reverse operations, that is raising a number to a power.

Let me just check your prior knowledge on squared and cubed numbers. Kindly,

list down the first 20 squared and cubed numbers on you notebook. This will help you to

get the roots of a real number.

All right learners! Based on the lists of squared and cubed numbers you wrote on

your notebook, make sure that those are all correct. Take a look at the list of square

roots and cubed roots of the first 30 positive integers below. You can compare your

answers by getting their roots.

Volume = 27 cm3

side = 3 cm


10

Here are the lists of the roots of real numbers of the first 20 positive integers.

Take time to analyze and study them thoroughly.

Lets have some examples of getting the roots of a real number.

SQUARE ROOTS

CUBE ROOTS


11

As you can see in the examples above, the roots of real numbers can be positive

or negative. Thats right! Every positive real number has two square roots: a positive

square root and a negative square root. For example, the square roots of 36 are 6 and

6, because and

A positive square root is also called the principal square root. The symbol is

called radical sign. The radical sign is used to denote the principal square root. To name

the negative root of a number, we use .

Thus, and . We can use the symbol to name the positive

and negative square roots.

Lets now find out if you really understood this lesson. Please answer the activity

below. Afterwards, ask the answer key from your facilitator to see how it went.

ACTIVITY

DIRECTIONS: Copy the questions and find the roots of the following:

1. 6.

2. 7.

3. 8.

4. 9.

5. 10.


12


After a thorough discussion of the lesson on finding the roots of a real

number, let us now summarize what you have learned.

Any expression with rational exponents represents a radical expression.

To find the roots of a real number means to find its square root, cube root,

fourth root, and so on depending on the index or order of the radical.

The roots of a real number can be a positive root, which is the principal

root and the negative root.

POSTTEST

DIRECTIONS: Copy the questions and find the roots of the following:

1. 6.

2. 7.

3. 8.

4. 9.

5. 10.

Answer the pretest below to see if you really understood the lesson. If you got a perfect

score, you may now take your mastery test. Otherwise, go back to the discussion part and

re-answer the activities.


13

LESSON 3

Simplifying Radical Expressions

Welcome dear learners! We are now on the third lesson of this module which is

all about simplifying radical expressions. Now that you have an idea on how to get the

roots of a real number, it will be helpful for you to simplify radical expressions.

There are some steps that you need to know on how to simplify radicals. How

would you know if an expression is already simplified? Simplifying radicals means that

the radicand does not contain any perfect powers of the index. The first step is to

express the radicand of the given radical as the product of two factors, one of which

must be a perfect nth power other than 1.

For instance, a square is in its simplest form if the radicand does not have a

factor which is a perfect square. Also, a cube root is in its simplest form if the radicand

does not have a factor which is a perfect cube.

Simplifying radical expressions is like solving your real-life problems. For you to solve it,

you need to plan carefully on the things that you need to settle. Then, you need to

simplify things instead of making them complicated. You need to have an exact solution

to your problems. After reading and studying this lesson, you will be able to simplify

radical expressions. Lets start by answering the pretest below. After which, check your

answers using the answer key.

PRETEST


1. 6.

2. 7.

3. 8.

4. 9.

5. 10.


14

Simplifying radical expressions is very simple, just follow the steps that I will

provide you. Like in simplifying exponents, theres also a rule in simplifying radicals that

should be considered. This law in simplifying radicals is important and helpful for you to

have an accurate answer and to make your solving easier.

The product rule for square roots is used to simplify radical expressions. It states

that the square root of the product is equal to the product of the square roots. We use

this rule to simplify radicals. But how would you know if a radical expression is already

in its simplified form? Take time to read and analyze the following conditions for you to

understand it very well.

A radical is said to be in its simplest form if the following conditions are satisfied:

1. The radicand has no factor raised to a power greater than or equal to the index;

2. The radicand has no factors;

3. No denominator contains a radical; and

4. Exponents in the radicand and the index of the radical have no common factor

except 1.

How do you find the conditions above? Dont worry, there is another technique for

you. To change radicals to their simplest form, you have to express the radicand into its

prime factors and then look for perfect nth powers in exponential form.

Lets have some examples on the next page for you to understand it thoroughly. This

time, we will apply the product rule and consider the conditions mentioned above to

simplify radicals.

PRODUCT RULE FOR RADICALS

If a and b are non negative numbers, then .


15

Examples: Simplify the following radical expressions:

a.

Apply the Product Rule, since the factors of 50 are

25 and 2.

Get the square root of 25 since it is a perfect

square.

Multiply and simplify.

b.

Apply the Product Rule since the factors of 48 are

16 and 3.

Get the square root of , then simplify.

Hint: To find the square root of , just divide the exponent by 2 since the index

is 2. Thus, .

c.

Apply the Product Rule since the factors of 72 are

36 and 2.

Get the square root of .

Simplify.


16

d.

Get the square root of since it is a perfect

square.

Hint: To find the square root of , just divide the exponent by 2 since the

index is 2. Thus, .

e.

Get the square root of since it is

a perfect square.

Hint: To find the square root of , just divide the exponent by 2 since the index

is 2. Thus, .

f.

Apply the Product Rule since the factors of

12 are 4 and 3.

Get the square root of

Simplify.

Hint: To find the square root of , just divide the exponent by 2 since the

index is 2. However, the exponent of x is 1 and it cant be divided exactly by 2. In

this case, x will remain inside the radical symbol. Thus, .


17

g.

Get the square root of since it is a

perfect square.

How do you find the examples above? Do you take into consideration the

conditions mentioned above? Or do you have another way of simplifying radical

expressions? Whatever it is that you have discovered, as long as you arrived at the

correct answer, go ahead.

Lets try harder examples for further understanding.

Examples: Simplify the following expressions:

h.

Apply the Product Rule; factors of 32 are 16

and 2.


Multiply the numbers outside the radical

sign, then simplify.

Hint: To find the square root of , divide the exponent by 2 since the index is

2. However, the exponent 5 cant be divided exactly by 2. So, you need to factor

, wherein the factors should contain an exponent that are divisible by 2. Thus

= . Therefore, .


18

i.

Apply the Product Rule. The factors of 300

are 100 and 3.

Get the square root of 100 which is 10.


symbol, 4 10 = 40.

j.

Apply the Product Rule. The factors of

are and .



sign, .

k.

Apply the Product Rule.

Get the square root then simplify.

l.


Get the square root then simplify.


19

ACITIVITY


1. 6.

2. 7.

3. 8.

4. 9.

5. 10.


After an in-depth discussion of the lesson on simplifying radical

expressions, we learned that:

Simplifying radicals means that the radicand is not to contain any perfect

powers of the index.

The product rule for square roots is used to simplify radical expressions. It

states that the square root of the product is equal to the product of the

square roots.

A radical is said to be in its simplest form if the following conditions are

satisfied.

The radicand has no factor raised to a power greater than or equal to the

index;

The radicand has no factors;

No denominator contains a radical; and

Exponents in the radicand and the index of the radical have no common

factor except 1.

Did you understand everything? Answer the activity below to apply what you have learned. After answering, get the answer key from your facilitator and check your work.


20

POSTTEST


1. 6.

2. 7.

3. 8.

4. 9.

5. 10.

Be ready to answer the posttest below. After which, compare

your answers with the answer key provided by your facilitator. Do your best and have fun answering the posttest!


21

LESSON 4

Addition and Subtraction of Radicals

Hello dear learners! We are now on the fourth lesson of this module which is all

about addition and subtraction of radical expressions. Adding and subtracting of radical

expressions is like adding and subtracting integers, however, you need to simplify the

radical expression first. Thus, the previous lesson would really be a big help for you to

perform the operations.

To add and subtract radicals is simply to combine like terms. For radicals to

become like terms, the index and the radicand must be the same. For instance,

and are like terms, while and are unlike terms. Thus, we

cannot combine an expression having unlike terms.

Adding and subtracting radical expressions is like how you treat your real-life

problems. You have to identify the main problem then think of ways on how you are

going to deal with it. You have to simplify it first before taking any action. Add more

patience and subtract negative vibes for a better result. Everything you do there has a

purpose, and that purpose is within you. Just figure it out.

After reading and studying this lesson, you will be able to add and subtract

radical expressions. Lets start with a pretest to check your prior knowledge about this

lesson. Please answer the pretest below then check your work using the answer key

provided by your facilitator.

PRETEST

DIRECTIONS: Simplify each of the following radical expressions.

1. 6.

2. 7.

3. 8.

4. 9.

5. 10.


22

Adding and subtracting radical expressions is simply combining like terms.

Having like terms means that the indices and the radicands are the same, otherwise

they are unlike. If the terms are unlike, you cant add or subtract radicals. Thus,

simplifying radicals will be used extensively in this lesson.

Look at the given examples below. Can you add or subtract the radical

expression? Why or why not?

Both examples cant be added or subtracted because the terms are unlike. In the

first example, , the radicands are different, thus you cant combine

them. Same with the second example, , though the radicals are the

same, the indices are different from each other. Thats why you cant also combine the

expressions.

Lets have another example wherein the terms are like. Analyze and observe

very carefully the examples below.

Example 1: Perform the indicated operation in the following radical expressions:

Solutions:

Distributive Property of Addition

Distributive Property of Addition


23

What did you notice in the previous examples? Thats right! The given radical

expressions are similar or like terms thats why we added and subtracted the radicals

easily.

Aside from having like terms, what else did you notice in the solution of the

above examples? Correct! To add and subtract radicals, we need to use the distributive

property of addition. This property will help you to simplify radical expressions.

Lets have more examples of addition and subtraction of radicals. Try to analyze

and observe the following examples.

Example 2: Simplify the following radical expressions:

Solutions:

Simplify each term. Use the Product Rule.

Get the square root.

Use the Distributive Property of Addition.

Simplify each term. Use the Product Rule.


Simplify.

Use the Distributive Property of Addition.


24

In the examples above, what have you observed? As you can see, the radicands

are not the same; so you cant add nor subtract the given radical expressions right

away. You have to simplify each term first so that you will have like terms. After that,

combine them. The product rule for radicals and the distributive property are also used

in adding and subtracting radicals.

Lets have more examples and try to figure out how it is done. Dont forget to use

the product rule and distributive property of addition to simplify radical expressions.

Example 3: Perform the indicated operations then simplify.

Solutions:

Get the square root of each term.

Simplify.

Get the square root of each term.

Simplify.


25

Use the Product Rule.


Simplify.

Use the Distributive Property

of Addition.

Apply Product Rule.


Simplify.

Combine similar terms.

Hint: Since the radicands are unlike, they cant be combined. Hence, the final

answer is .

How is it going? Did you fully understand the lesson? I hope that the examples

were sufficient. Answer the activities on the next page and check your work with the

answer key.


26

ACITIVITY

DIRECTIONS: Perform the indicated operations then simplify.

1. 6.

2. 7.

3. 8.

4. 9.

5. 10.


After a thorough discussion of the lesson on addition and subtraction of

radical expressions, we learned that:

To add and subtract radicals is simply to combine like terms wherein the

radicands and indices are the same.

In adding and subtracting radicals, the product rule for radicals is also

needed.

Simplifying radicals having unlike terms is helpful in adding and

subtracting radical expressions.

Moreover, the distributive property of addition is also important in adding

and subtracting radicals.

Be ready to answer the posttest below. After which, compare your answers with the answer key

provided by your facilitator. Do your best and have fun answering the posttest!


27

POSTTEST


1. 6.

2. 7.

3. 8.

4. 9.

5. 10.


28

LESSON 5

Multiplication and Division of Radicals

Hello learners! Youre doing great! You are now on the fifth lesson of this module.

Lets continue our journey on operations of radicals. The skills and techniques that you

learned in the previous lesson will be helpful in understanding our new lesson. This is

just the continuation of what you have started in Lesson 4 of this module.

How to multiply and divide radicals? Multiplying radicals is basically the same as

multiplying algebraic expressions wherein the distributive property of multiplication is

needed. However, in multiplying radicals, there are some cases you need to consider

which will be discussed on this lesson. On the other hand, dividing radicals is almost the

same as simplifying radicals. If you could still remember in our second lesson, one of

the considerations is that no denominator contains a radical. The process of removing

the radicals in the denominator is called rationalization.

Do you want to learn more? Just continue reading this module and youll learn a

lot. But before we proceed, answer the pretest to see your prior knowledge on this

lesson. Afterwards, ask your facilitator for the answer key so you can check your work.

PRETEST


1. 6.

2. 7.

3. 8.

4. 9.

5. 10.


29

MULTIPLICATION OF RADICALS

Were almost in the last lesson of this module and its getting harder, dont you

think? Operations on radicals are not easy, especially the lesson on multiplication and

division of radicals. Youll need all the skills you acquired and mastered in this module

and in the previous modules. It would be easy for you to do this lesson if you still

remember all those skills.

To multiply a monomial to a monomial expression involving radicals, the indices

should be the same for you to multiply the radicands. Moreover, recall the use of the

distributive property when finding the product of a monomial and a polynomial. In a

similar manner, the distributive property provides the basis of finding certain special

products that involve radicals. Distributive property of multiplication is used if you are

multiplying a monomial and a polynomial. In addition, we can multiply binomial

expressions involving radicals by using the FOIL method. When the binomial is

squared, we apply the process we use in squaring algebraic expressions.

Try to recall the skills you learned in the previous modules because they will help

you a lot to understand this lesson. Lets have some examples for further

understanding.

Example 1: Find the product of the following radical expressions:

a.

Multiply the radicands.


Apply the Product Rule for radicals.


30

b.

Multiply the whole numbers then multiply the

radicals.

Get the square root of 108.


Simplify.

What did you notice in the previous examples? Isnt that easy? Of course!

Multiplying radicals is easy, if and only if you mastered the skills in the previous lessons.

To multiply a monomial to a monomial expression involving radicals is similar to

multiplying algebraic expressions. However, the indices should be the same for you to

multiply the radicands. Then after multiplying the radicands, simplify the radical

expression. Moreover, the product rule for radicals is also used.

Lets now proceed to more challenging examples. Carefully study the steps in

multiplying a monomial to a binomial expression involving radicals.


c.

Use the distributive property of multiplication.

Distribute to the number inside the parenthesis.


Get the factors of only since is a prime.

Get the square root of 4.

Hint: Since the radicands are different, we cannot combine the radicals; hence,

the final answer is .


31

d.

Use the Distributive Property.

Distribute to the number inside the parenthesis.

Multiply the whole numbers then multiply the

radicands.

Hint: Since the radicands are different, we cant combine the radicals; hence, the

final answer is .

How do you find multiplying a monomial to a binomial expression involving

radicals? As youve noticed, the distributive property of multiplication is very helpful. To

distribute means to multiply. Also, the product rule for radicals is used in this lesson.

Simplifying your answer is really important in this lesson. So you better master the

lessons on how to simplify radical expressions. The concept of adding and subtracting

radicals is also needed in this lesson. Remember, you can only combine radicals if they

are like terms.

Moving on, what if you are multiplying a binomial to a binomial expression

involving radicals? To do this, we will use the FOIL method. If you still remember, we

used the FOIL method in our lesson on special products. This method can only be used

if you are multiplying a binomial to a binomial expression.

Lets recall special products. For instance, if you are looking for the product of

and , just apply the FOIL method.

Last

Inner

Outer

First FirstOuterInnerLast

Combine

similar terms.


32

In addition, when the binomial is squared, we also apply the process we use in

squaring algebraic expressions.

Moreover, when multiplying the sum and difference of two binomials, we also use

the same method.

Lets have some examples of multiplying a binomial to a binomial expression

involving radical expressions.


e.

Use the FOIL method.


Thus, .

Hint: Since the radicands are different from each other, we cannot combine

them.

Squaring a Binomial

and

Sum and Difference of 2 Binomials


33

f.

Use the FOIL method.

Simplify: .

Multiply: .


g.

Square of a binomial,

.

Simplify.


Multiply then combine similar

terms.

h.

Square of a binomial,

.

Simplify.


Multiply then combine similar

terms.

Did you understand all the examples illustrated above? Turn to the next page to

learn how to multiply other cases of radical expressions.


34

A. INDICES ARE DIFFERENT BUT RADICANDS ARE THE SAME

To find the product of radicals with different indices but the same radicands,

look at the examples below and follow the given steps.

Examples:

a. 3 55

= 31

2

1

55 (Transform the radicals to powers with fractional

exponents)

= 31

2

1

5 (Add the powers)

= 65

5 (Simplify)

= 6 55 (Rewrite the product as a single radical then simplify)

= 6 3125

b. 43 222

= 41

3

1

2

1

222

= 41

3

1

2

1

2

= 1213

2

= 12 132

= 12 12 22

= 1212 12 22

= 12 22


35

B. INDICES AND RADICANDS ARE DIFFERENT

To find the product of radicals with different indices and radicands, study

the examples and follow the given steps.

Examples:

a. 3 42

= 31

2

1

42 (Transform the radicals to powers with fractional

exponents)

= 62

6

3

42 (Change the fractional exponents into similar

fractions)

= 6 23 42 (Rewrite the product as a single radical)

= 6 168 (Simplify)

= 6 128

= 6 264

= 66 6 22

= 6 22

b. 324

= 21

4

1

32

= 42

4

1

32

= 4 232

= 4 92

= 4 18


36

c. 63 632

= 61

3

1

2

1

632

= 61

6

2

6

3

632

= 6 23 632

= 6 698

= 6 432

Did you understand the concepts on multiplication of radicals? We will now go to

the second part of this lesson which is division of radicals.

Dividing radicals includes addition, subtraction, and multiplication of radicals.

Thats right learners! All the skills that you have learned in the previous lesson will be

used in dividing radicals. How does it sound to you? Just apply all the rules and laws

presented to make solving easier.

DIVISION OF RADICALS

Dividing radicals is almost the same as simplifying radicals and dividing algebraic

expressions. For instance, if you divide by , the quotient would be . Isnt

that easy?

Let try to divide the following radical expressions. Study the examples below.

Examples: Divide the following radical expressions then simplify.

a.

b.


37

Dividing radicals is simply dividing the radicands, that is if it is possible to divide,

as shown in example a. In example b, the expression was divided and then simplified.

Since is a perfect square, you have to get its square root and multiply it to the whole

number.

Theres another case of dividing radical expressions. If you could still remember

in our first lesson, one of the considerations is that no denominator contains a radical. If

you notice in the example above, theres no denominator that contains a radical. It

means that it is already simplified. The process of removing the radicals in the

denominator is called rationalization.

For further understanding of what rationalizing is, lets have some examples and

try to analyze how it is done.

Examples: Rationalize the following radical expressions.

c.

Multiply the fraction by since the denominator is

Get the square root of the denominator.

d.

Multiply the fraction by since the denominator is

Get the square root of the denominator.

Reduce the fraction to its lowest term, .


38

e.

Multiply the fraction by since the denominator with

radical is .

Get the square root of the denominator, .

Multiply the denominator.

Apply the Product Rule in the numerator, then get the

square root of .

Reduce the fraction to its lowest term, .

How did you find the previous examples? Its a bit harder compared to the first

example of division of radicals. Rationalizing a radical expression simply means

simplifying. To simplify the expression, you have to remove the radicals in the

denominator. Simply multiply the fraction, both numerator and denominator, to the

radical expression in the denominator.

Also, as you noticed, the denominators in the previous examples are all

monomial. What if the expression in the denominator is a binomial? Is it the same

rationalizing? Lets find out by studying the following examples.


39

Example: Simplify the following radical expression:

f.

Get the conjugate of the denominator which is

, then multiply it to both numerator and

denominator.

Use the distributive property in the numerator,

then apply the FOIL method in the denominator .

Simplify the denominator by getting the square root,

Subtract the denominators.

Since there are no like terms in the numerator, you cannot combine them.

Before we proceed to the next examples, lets find out first what a conjugate is.

As we have stated in the previous lesson, a simplified radical expression should have

no radical in the denominator. Whenever a radical expression has a binomial with

square root radicals in the denominator, rationalize the denominator by multiplying both

the numerator and the denominator by the conjugate of the denominator. The conjugate

of is and the conjugate of is .


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g.

Get the conjugate of which is ,

then multiply it by the numerator and

denominator.

Use the distribute property in the numerator,

then use the FOIL method in the denominator.

Simplify the denominator by getting the

square root, i.e. .

Simplify by subtracting the denominator.


h.

Get the conjugate of which is ,

then multiply it by the numerator and denominator.

Use the FOIL method on both numerator and

denominator.

Apply the product rule, i.e. .

Simplify the denominator by getting their square

roots.

Simplify the denominator by subtracting them.


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ACITIVITY

A. DIRECTIONS: Multiply the following radical expressions then simplify.

1. 7.

2. 8.

3. 9.

4. 10.

5. 11.

6. 12.

B. DIRECTIONS: Divide the following radical expressions then simplify.

1. 6.

2. 7.

3. 8.

4. 9.

5. 10.

Did you understand the discussion? Answer the activity below and apply everything youve learned. After answering, get the answer key from your facilitator to find out your score.


42


After a thorough discussion of the lesson on multiplying and diving radical

expressions, we learned that:

To multiply radicals, simply get the product of the radicands if and only if the

indices are the same. The distributive property is also used in finding the

product of a radical expression.

We can multiply binomial expressions involving radicals by using the FOIL

method.

Simplifying radicals means that no denominator contains a radical.

Dividing radicals is simply removing the radicals in the denominator. This

process is called rationalization.

Whenever a radical expression has a binomial with square root radicals in the

denominator, rationalize the denominator by multiplying both the numerator

and the denominator by the conjugate of the denominator. The conjugate of

is and the conjugate of is .

Answer the posttest on the next page

then check your answers with the answer key. If you get a perfect score, take your mastery test. Otherwise,

revisit the discussion part and re-answer the activities.


43

POSTTEST

A. DIRECTIONS: Multiply the following radical expressions then simplify.

1. 7.

2. 8.

3. 9.

4. 10.

5. 11.

6. 12.

B. DIRECTIONS: Rationalize the following radical expressions then simplify.

1. 6.

2. 7.

3. 8.

4. 9.

5. 10.


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LESSON 6

Radical Equations

Finally! Were now on the last lesson of this module. In this lesson, you will solve

unknown variables involving radicals, called radical equations. What is a radical

equation? We often refer to an equation that contains radicals with variables in a

radicand as radical equations. To solve a radical equation, we need to use the power

rule. It states that if both sides of an equation are raised to the same power, all solutions

of the original equation are also solutions of the new equation.

There are some steps in solving a radical equation that you need to consider: (a)

Isolate the radical. Make sure that one radical term is alone on one side of the equation;

(b) Apply the power rule. Raise each side of the equation to a power that is the same as

the index of the radical; (c) Solve the resulting equation. If it still contains a radical,

repeat steps 1 and 2; (d) Check to determine if the roots of the resulting equation are

roots of the original equation. If not, reject any such roots as an extraneous value.

Lets start by answering the pretest below. After answering, use the answer key

provided by your facilitator to check your work. Have fun!

PRETEST

DIRECTIONS: Solve the following radical equations.

1.

2.

3.

4.

5.


45

What is a radical equation? A radical equation is an equation that contains

radicals with variables in a radicand. To solve a radical equation, we need to use the

power rule. It states that if both sides of an equation are raised to the same power, all

solutions of the original equation are also solutions of the new equation.

In this lesson, we will solve equations that contain square roots of variable

expressions. To solve such, we will use the square root property that we studied

quadratic equations. Square root property states that if both sides of an equation are

squared, all solutions of the original equations are solutions of the new solution.

There are some steps in solving a radical equation that you need to consider.

1. Isolate the radical. Make sure that one radical term is alone on one

side of the equation.

2. Apply the power rule. Raise each side of the equation to a power that

is the same as the index of the radical.

3. Solve the resulting equation. If it still contains a radical, repeat steps a

and b.

4. Check to determine if the roots of the resulting equation are roots of

the original equation. If not, reject any such roots as an extraneous

value.

Lets now have some examples to illustrate this. Study and analyze them carefully.

Example: Solve.

1.

Apply square root property on both sides.

Use the power rule.

Transpose 4 to the right side.

Subtract 64 and 4.

Divide both sides by 3 to solve x.


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2.

Apply the square root property.

Use the power rule.


Simplify.

Divide both sides by 2 to solve y

3.

Isolate the radical and transpose 10 to the

right side.


Use the power rule.

Transpose 1 to combine similar terms.

Simplify.

Divide both sides by 9 to solve w.

4.


Use the power rule.

Use the distributive property.


Simplify.

Divide both sides by 4 to solve b.


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How did you find the examples above? Just follow the steps in solving radical

equations and everything will be easy to solve. Dont forget to apply all the rules and

properties you have learned from the previous lessons and modules. Let me check if

you really understood the lesson. Answer the activity below. Afterwards, check your

answers with the answer key provided by your facilitator.

ACTIVITY


1.

2.

3.

4.

5.


After a thorough discussion of the lesson on solving radical equations, let

us now summarize what you have learned.

A radical equation is an equation that contains radicals with variables

in a radicand.

The power rule states that if both sides of an equation are raised to the

same power, all solutions of the original equation are also solutions of

the new equation.

The square root property states that if both sides of an equation are

squared, all solutions of the original equations are solutions of the new

solution.


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POSTTEST


1.

2.

3.

4.

5.

You are now on the last step of this module. Answer the posttest below then check your answers. If you got a perfect score, you may now

take the mastery test. Otherwise, read again the discussion part and re-answer the activities. Good

luck!


49

REFERENCES

Alferez, M. S. and Duro, M.C. (2004). MSA Elementary Algebra. Gerpress Printing:

Quezon City.

Alferez, M. S. and Duro, M.C. (2004). MSA Intermediate Algebra. GerpressPrinting:

Quezon City.

Aoanan, G. O. et. al. (2011). Next Generation Math II. Diwa Learning Systems, Inc:

Makati City.

Bautista, E. P. et al. (2006). XP Intermediate Algebra II. Vibal Publishing House, Inc:

Quezon City.

Isidro, C. D.(1999). Elementary Algebra for High School II. Anvil Publishing, Inc: Pasig

City.

Documents

MATH 10 Module 4 (Rational Exponents and Radical Expressions)