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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
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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.
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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.
Example 2: Simplify the following expressions.
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.
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Example 3: Simplify the following expressions.
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.
Example 4: Simplify the following 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.
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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!
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POSTTEST
DIRECTIONS: Simplify each of the following expressions.
1.
2.
3.
4.
5.
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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
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PRETEST
DIRECTIONS: Simplify each of the following expressions.
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
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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
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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
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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.
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What have you learned so far?
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.
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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
DIRECTIONS: Simplify each of the following expressions.
1. 6.
2. 7.
3. 8.
4. 9.
5. 10.
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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 .
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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.
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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, .
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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.
Get the square root of .
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, .
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i.
Apply the Product Rule. The factors of 300
are 100 and 3.
Get the square root of 100 which is 10.
Multiply the numbers outside the radical
symbol, 4 10 = 40.
j.
Apply the Product Rule. The factors of
are and .
Get the square root of .
Multiply the numbers outside the radical
sign, .
k.
Apply the Product Rule.
Get the square root then simplify.
l.
Apply the Product Rule.
Get the square root then simplify.
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ACITIVITY
DIRECTIONS: Simplify each of the following expressions.
1. 6.
2. 7.
3. 8.
4. 9.
5. 10.
What have you learned so far?
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.
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POSTTEST
DIRECTIONS: Simplify each of the following expressions.
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!
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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.
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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
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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.
Get the square root.
Simplify.
Use the Distributive Property of Addition.
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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.
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Use the Product Rule.
Get the square root.
Simplify.
Use the Distributive Property
of Addition.
Apply Product Rule.
Get the square root.
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.
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ACITIVITY
DIRECTIONS: Perform the indicated operations then simplify.
1. 6.
2. 7.
3. 8.
4. 9.
5. 10.
What have you learned so far?
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!
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POSTTEST
DIRECTIONS: Perform the indicated operations then simplify.
1. 6.
2. 7.
3. 8.
4. 9.
5. 10.
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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
DIRECTIONS: Perform the indicated operations then simplify.
1. 6.
2. 7.
3. 8.
4. 9.
5. 10.
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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.
Get the square root.
Apply the Product Rule for radicals.
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b.
Multiply the whole numbers then multiply the
radicals.
Get the square root of 108.
Apply the Product Rule for radicals.
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.
Example 2: Find the product of the following radical expressions:
c.
Use the distributive property of multiplication.
Distribute to the number inside the parenthesis.
Apply the Product Rule for radicals.
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 .
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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.
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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.
Example 3: Find the product of the following radical expressions:
e.
Use the FOIL method.
Apply the Product Rule.
Thus, .
Hint: Since the radicands are different from each other, we cannot combine
them.
Squaring a Binomial
and
Sum and Difference of 2 Binomials
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f.
Use the FOIL method.
Simplify: .
Multiply: .
Combine similar terms.
g.
Square of a binomial,
.
Simplify.
Get the square root.
Multiply then combine similar
terms.
h.
Square of a binomial,
.
Simplify.
Get the square root.
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.
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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
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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
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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.
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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, .
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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.
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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.
Since there are no like terms in the numerator, you cannot combine them.
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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Since there are no like terms in the numerator, you cannot combine them.
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.
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What have you learned so far?
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.
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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.
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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.
Combine similar terms.
Simplify.
Divide both sides by 2 to solve y
3.
Isolate the radical and transpose 10 to the
right side.
Apply the square root property.
Use the power rule.
Transpose 1 to combine similar terms.
Simplify.
Divide both sides by 9 to solve w.
4.
Apply the square root property.
Use the power rule.
Use the distributive property.
Combine similar terms.
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
DIRECTIONS: Solve the following radical equations.
1.
2.
3.
4.
5.
What have you learned so far?
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
DIRECTIONS: Solve the following radical equations.
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!
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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.