Download Math 9th grade LEARNING OBJECT Exploring exponents and

Math 9th grade
LEARNING UNIT
A special number
complex numbers
S/K
LEARNING OBJECT
Exploring exponents and radicals in real numbers
set:
SCO 1: Recognize the concept of exponent.
Skill 1: Determine exponents as the expression
of a product of equal factors.
Skill 2: Find positive exponents of real numbers.
Skill 3: Find negative exponents of real numbers.
Skill 4: Conjecture about the result of the
negative and positive exponents.
Skill 5: Find rational exponents of real numbers.
Skill 6: Interpret information on problems
associated with exponents.
SCO 2: Identify the properties of exponents.
Skill 1: Deduct the properties exponents through
generalizations of cases and patterns.
Skill 2: Apply the properties of exponents to
simplify algebraic expressions.
SCO 3: Recognize the concept of radicals.
Skill 1: Recognize the radicals as an expression
that helps finding the base of an exponent.
Skill 2: Find the even roots for real numbers.
Skill 3: Find the odd roots for real numbers.
Skill 4: Determine the roots that have no real
solution.
SCO 4: Identify the properties of radicals.
Skill 1: Deduct the properties of radicals based
on the relation with exponents.
Skill 2: Apply the properties of exponents to
simplify algebraic expressions.
Skill 3: Simplify radicals with the use of
properties.
Skill 4: Simplify algebraic expressions using
properties of exponents and radicals.
Skill 5: Justify using properties of radicals to
simplify expressions.
SCO 5: Identify operations with radicals.
Skill 1: Apply properties of real numbers and
roots to add and subtract radicals.
Skill 2: Multiply radicals with a common index.
Skill 3: Multiply radicals with a different index by
reducing them to a common index.
Skill 4: Divide radicals.
SCO 6: Rationalize algebraic fractions.
Skill 1: Identify the rationalization of algebraic
fractions.
Skill 2: Rationalize algebraic fractions with
monomial denominators.
Skill 3: Rationalize algebraic fractions with
binomial denominators.
Language
English
Socio cultural context of School, house.
the LO
Curricular axis
Numerical thinking and numerical systems.
Standard competencies
Identify the use of exponents, radicals, and logarithms
to represent mathematical and non-mathematical
situations, and to solve problems.
Background Knowledge
Basic operations with real numbers, simplifying similar
terms, and algebraic fractions.
Basic Learning
Recognize the meaning of the positive and negative
Rights
rational exponents using the law of exponents.
Vocabulary box
Once Upon A Time, Subjects, Cheer Him Up, Array,
Undermined, Delay
English Review Topic
Present Tense, Adjectives
Vocabulary box
Once upon a time: a long time ago.
Subjects: people that work for a sultan; commonly
slaves
Cheer him up: To improve someone´s mood
Array: large group of things
Undermined: to give little or no importance
Delay: to extend time, to postpone
NAME: _________________________________________________
GRADE: ________________________________________________
INTRODUCTION
There is a story about the origin of chess in a mathematical context, this
is the story:
Once upon a time there was a king named Sheram who was depressed
after the loss of his son, nothing seemed to console him. His subjects tried
to cheer him up with games, jokes, and an array of activities but none of
them worked.
Sissa, one of his subjects who had also lost a son, came upon his king to
show him a new game and how to play it: the game was chess. After
several games the king had recovered his happiness and offered Sissa
anything he wanted, he was a powerful king so he could do it. Sissa
thought about it and after a few moments his request was:
For each square on the chess board he requested wheat. 1 grain of wheat
for the first square, 2 for the second, 4 for the third, 8 for the fourth and
son on; each square should have twice as the previous one.
The King felt as if his power was being undermined and agreed with a
frown. Sissa was asked to leave, and the ministers would pay the debt.
After some time, the king asked about the total amount of wheat to be
delivered to Sissa, his subjects did not have an answer as the best
mathematicians of the kingdoms had not solved it. The king could not
understand the delay and asked for a detailed report. The report explained
that the amount of wheat requested by Sissa was bigger than the amount
of wheat in the kingdom or even the world.
What amount of wheat must be payed to Sissa?
The development of this learning object will help us answer this question.
Objectives
-
To recognize the properties of exponents and radicals in real
numbers.
To determine the definition of exponents.
To use the properties of exponents to simplify algebraic
expressions.
To determine the definition of radicals.
To use the properties of radicals to simplify algebraic expressions.
To simplify radicals and define operations between them.
To determine the process to rationalize algebraic fractions.
Activity 1
SKILLS:
Skill 1: Determine exponents as the expression of a product of equal
factors.
Skill 2: Find positive exponents of real numbers.
Skill 3: Find negative exponents of real numbers.
Skill 4: Conjecture about the result of the negative and positive
exponents.
Skill 5: Find rational exponents of real numbers.
Skill 6: Interpret information on problems associated with exponents.
Skill 1: Deduct the properties exponents through generalizations of cases
and patterns.
Skill 2: Apply the properties of exponents to simplify algebraic
expressions.
The story in the introduction can be solved using exponents with base 2.
As the story told, the first square of the chess board will have 1 grain, the
next one 2, the third one 4 and so on; each square should have twice as
the previous one. To answer this question, it is necessary to review some
of the concepts and definitions of exponents, as well as some of the
properties.
Exponents
Exponents are expressed with numbers a and n, then the nth power of
a is:
𝑎𝑛 = ⏟
𝑎 ∗𝑎 ∗𝑎 ∗𝑎 ∗ …∗ 𝑎 = 𝑐
𝑛 𝑓𝑎𝑐𝑡𝑜𝑟𝑠
The number a is called the base, n is called the exponent, and c is called
the power. In other words, “a” is multiplied “n times”.
It is evident that 𝑎1 = 𝑎; and by definition 𝑎0 = 1.
Based on the previous information, the king´s question can be answered
as follows:
A chess board has 64 squares; the first grain in the first square will be
represented like this:
1 = 20
Why is the base 2? Because the result is doubled every next square.
The second square has twice the first square, its representation is:
2 = 21
The third square has 4 wheat grains and its representation is:
4 = 22
The process repeats itself and the results are in the following table:
Square
Wheat grains
1 = 20
1
2 = 21
2
4 = 22
3
8 = 23
4
⋮
⋮
263
64
Finally, the amount of wheat the King has to pay Sissa can be found as
follows:
20 + 21 + 22 + 23 + ⋯ + 263
Using a calculator, it is possible to find the value of the expression and
determine the debt cannot be paid after many years in which the country
produced that amount of wheat.
Consider this…
At the beginning when the definition of exponents was given, the set of
numbers they belong to was not defined on purpose. 𝑎, 𝑏 and 𝑐 belong to
which number set?
𝑎, 𝑏 and 𝑐 ∈ 𝑅,
1
If so, how can a number have − 2 factors? Exponents have several
properties that will help answer this type of questions.
Laws of exponents
The following laws are easy to deduct if the definition of exponents has
been understood and they are necessary to simplify algebraic
expressions:
Consider the product 𝑥 2 ∗ 𝑥 3 ; use the definition of exponents to solve:
𝑥 2 ∗ 𝑥 3 = (𝑥 ∗ 𝑥) ∗ (𝑥 ∗ 𝑥 ∗ 𝑥) = 𝑥 ∗ 𝑥 ∗ 𝑥 ∗ 𝑥 ∗ 𝑥 = 𝑥 5
How do the first exponents and the last one relate? The resulting exponent
is the sum of the first two.
Using the same logic, we can generalize the procedure, thus finding the
first law:
𝒂𝒃 ∗ 𝒂𝒄 = 𝒂𝒃+𝒄
Using the first law, it we can see what happens in the division. Consider
𝑦5
the example 𝑦 3; expand both denominator and numerator to solve:
𝑦5 𝑦 ∗ 𝑦 ∗ 𝑦 ∗ 𝑦 ∗ 𝑦 𝑦 ∗ 𝑦 ∗ 𝑦
=
=
∗ 𝑦 ∗ 𝑦 = 1 ∗ 𝑦 ∗ 𝑦 = 𝑦 ∗ 𝑦 = 𝑦2
𝑦3
𝑦∗𝑦∗𝑦
𝑦∗𝑦∗𝑦
How do the first exponents and the last one relate? The resulting exponent
is the sum of the first two, resulting in the second law of exponents.
Consider the same exercise in which the exponents are the same. Using
the same logic as before, the result is 1.
𝒂𝒃
= 𝒂𝒃−𝒄 ; 𝒂𝒄 ≠ 𝟎
𝒂𝒄
Now consider: (𝑥 ∗ 𝑦)4 . Develop the algebraic expression using the
definition of exponents:
(𝑥 ∗ 𝑦)4 = (𝑥 ∗ 𝑦) ∗ (𝑥 ∗ 𝑦) ∗ (𝑥 ∗ 𝑦) ∗ (𝑥 ∗ 𝑦)
=𝑥∗𝑦∗𝑥∗𝑦∗𝑥∗𝑦∗𝑥∗𝑦
=𝑥∗𝑥∗𝑥∗𝑥∗𝑦∗𝑦∗𝑦∗𝑦
= 𝑥4 ∗ 𝑦4
Based on the previous process it is possible to conclude the third law:
(𝒂 ∗ 𝒃)𝒄 = 𝒂𝒄 ∗ 𝒃𝒄
Following the same process as with the third law, it is possible to conclude
𝑧 3
the fourth property. Use (𝑤) as an example:
𝑧 3
𝑧
𝑧
𝑧
( ) =( )∗( )∗( )
𝑤
𝑤
𝑤
𝑤
=
𝑧∗𝑧∗𝑧
𝑧3
= 3
𝑤∗𝑤∗𝑤 𝑤
Therefore, the exponent will be distributed in the division, allowing us to
reach the fourth law:
𝒂 𝒄 𝒂𝒄
( ) = 𝒄 ; 𝒃𝒄 ≠ 𝟎
𝒃
𝒃
Consider another example: (𝑥 3 )2 . Using the definition of exponents, solve:
(𝑥 3 )2 = 𝑥 3 ∗ 𝑥 3 = (𝑥 ∗ 𝑥 ∗ 𝑥) ∗ (𝑥 ∗ 𝑥 ∗ 𝑥))
= 𝑥 ∗ 𝑥 ∗ 𝑥 ∗ 𝑥 ∗ 𝑥 ∗ 𝑥 = 𝑥6
How do the first exponents and the last one relate? The resulting exponent
is the product of the first two, resulting in the fifth law of exponents. The
power of another power results in the product of the exponents:
𝒄
(𝒂𝒃 ) = 𝒂𝒃∗𝒄
All of the previous laws have been determined using the law of exponents
and some multiplication properties since the power is the result of
multiplying the same factor n times. But we have not answered the
1
question yet: how can a number have − 2 factors? Or even so, how do we
solve a factor with a negative or rational exponent?
The following laws will only be explained but not proven due to the level
of complexity and they will be used to answer the initial question.
Sixth law of exponents:
𝒂−𝒃 =
𝟏
; 𝒂𝒃 ≠ 𝟎
𝒂𝒃
This law states that a power with a negative exponent can be rewritten
as the reciprocal with the positive exponent. As seen in the example:
5−2 =
1
1
=
2
5
25
This law allows solving powers with negative exponents.
To raise a fraction to a negative power, invert the fraction and change the
sign of the exponent:
𝒂 −𝒄
𝒃 𝒄
( ) =( )
𝒃
𝒂
To move a number raised to a power from the numerator to the
denominator or vice versa, change the sign of the exponent (seventh
law).
𝟏
𝒂 −𝒄 𝒂−𝒄 𝒂𝒄 𝒃𝒄
𝒃 𝒄
( ) = −𝒄 =
= 𝒄=( )
𝟏
𝒃
𝒃
𝒂
𝒂
𝒃𝒄
For rational exponents we have the eighth law:
𝒃
𝟏⁄
𝒃
√𝒂 = 𝒂
This law will be explained later because it requires an operation that has
not been introduced.
The following examples will show how the laws of exponents can be
applied to simplify algebraic expressions:
Example 1:
1 4 −3 6 𝑛 1
2 11 𝑛
2 𝑎 𝑏 𝑐 𝑑 = 2 𝑎4−2 𝑏 −3−4 𝑐 6−(−5) 𝑑 𝑛 = 4 𝑎2 𝑏 −7 𝑐 11 𝑑𝑛 = 2𝑎 𝑐 𝑑
3 2 4 −5
3
6
3𝑏 7
𝑎
𝑏
𝑐
4
4
Example 2:
𝑥𝑦𝑧 0
) =1
𝑎3 𝑏 2 𝑐 4
(
Learning activity 1
1. Simplify the algebraic expressions using the laws of exponents:
a. (((1010 )6 )15 )0
b.
(25)3 125−2 (625)−3
1 2
5
( ) 56
c.
6 −2 3 4
𝑎 𝑏 𝑐
7
5 1 5 −15
𝑎 𝑏 𝑐
3
𝑥 8 𝑦 0 𝑧 −3
−6
d. (27𝑥 2 𝑦 −4 𝑧 2 )
SKILL
2. Share your results with your classmates.
Activity 2
Skill 1: Recognize the radicals as an expression that helps finding the base
of an exponent.
Skill 2: Find the even roots for real numbers.
Skill 3: Find the odd roots for real numbers.
Skill 4: Determine the roots that have no real solution.
Skill 1: Deduct the properties of radicals based on the relation with
exponents.
Skill 3: Simplify radicals with the use of properties.
Skill 5: Justify using properties of radicals to simplify expressions.
Skill 1: Apply properties of real numbers and roots to add and subtract
radicals.
Skill 2: Multiply radicals with a common index.
Skill 3: Multiply radicals with a different index by reducing them to a
common index.
Skill 4: Divide radicals.
Skill 4: Simplify algebraic expressions using properties of exponents and
radicals.
What are radicals?
Radicals are similar to exponents; evidenced in the following formula:
𝑛
√𝑐 = 𝑎 ↔ 𝑎 𝑛 = 𝑐
𝑎 is the 𝑛𝑡ℎ 𝑟𝑜𝑜𝑡 of 𝑐 if and only if 𝑎 to the power of 𝑏 equals 𝑐.
The root of 𝑛 is known as the radical index, 𝑎 is the root and 𝑐 is the
radicand. By knowing the values of n and c it is possible to find a.
Therefore, if the power of a number is known, so is its root. For example,
5
what is √32?
To solve it must be written as an equation:
5
√32 = 𝑦
Using the definition of radicals:
𝑦 5 = 32
This equation can be understood as follows: what number multiplied by
itself 5 times equals 32? We can find the answer by trial and error. 1 is
not the answer because:
1 ∗ 1 ∗ 1 ∗ 1 ∗ 1 = 1, but 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 = 32. Therefore, 25 = 32, and
𝑦 = 2, then √32 = 2.
5
Using the same process, what is the value of √−16?
2
Written as an equation:
2
√−16 = 𝑦
By definition of radicals:
𝑦 2 = −16
What number multiplied by itself two times yields -16? No number in the
real set of numbers will answer this question. A negative number
multiplied by itself twice becomes positive, that is, the product of two
negative numbers is positive, the same as the product of two positive
numbers.
There is no real set of numbers that multiplied by itself will yield -16,
2
meaning that √−16 has no solution or the solution is not a real number.
Is it the same for other roots? No, the following examples will show which
negative roots exist and which do not in the set of real numbers.
Examples:
If we want to determine the value of √−1, that is, a number that multiplied
by itself six times yields -1; such number does not exist:
6
(−1) ∗ (−1) ∗ (−1) ∗ (−1) ∗ (−1) ∗ (−1) = 1 and
(1) ∗ (1) ∗ (1) ∗ (1) ∗ (1) ∗ (1) = 1
To find the value of √−8, write the equation:
3
3
√−8 = 𝑦
Then,
𝑦 3 = −8
Which number multiplied by itself three times yields -8? The answer is 2:
(−2)3 = −8
This is a real solution (belongs to the set of real numbers)
3
√−8 = −2
Even indexes cannot have negative roots, whilst odd indexes can have
negative roots.
The even index must have a positive root to have a radicand in the set of
real numbers because the product of two negative numbers is always
positive. If the number of factors is even and the root is negative, the
result is positive.
To draw another conclusion, it is necessary to give an example:
Solve √625.
4
Write an equation:
4
√625 = 𝑦
Then,
𝑦 4 = 625
What number multiplied by itself 4 times yields 625? There are two
possible answers, 5 and -5 since:
5 ∗ 5 ∗ 5 ∗ 5 = 625 and (−5) ∗ (−5) ∗ (−5) ∗ (−5) = 625.
Therefore,
4
√625 = ±5
Repeating the exercise with other even indexes the result is the same,
two roots are given, one positive and one negative.
Another example is √16 = ±4.
2
The odd index root of a negative radicand will yield a negative root as
seen before, such root is negative and is part of the set of real numbers:
3
√−8 = −2.
Properties of roots
1
The eighth law of exponents, √𝑎 = 𝑎 ⁄𝑏 , allows a root to be written as an
exponent and vice versa. This allows the use of the laws of exponents to
be used in radicals.
𝑏
Before continuing it is important to know that √𝑎 = 𝑎
understand:
𝑏
𝑏
𝑏
1⁄
𝑏
𝑏
√𝑎𝑏 = ( √𝑎) = 𝑎.
A practical result that allows relating exponents and radicals.
1. Distributive property for multiplying radicals
Expresses as:
helps to
𝒄
𝒄
𝒄
√𝒂 ∗ 𝒃 = √
𝒂 ∗ √𝒃
Results from using the third law of exponents:
(𝑎 ∗ 𝑏)𝑐 = 𝑎𝑐 ∗ 𝑏 𝑐
Like this:
𝑐
√𝑎 ∗ 𝑏 = (𝑎 ∗ 𝑏)
1⁄
𝑐
=𝑎
1⁄
𝑐
∗𝑏
1⁄
𝑐
𝑐
𝑐
= √𝑎 ∗ √𝑏
This property is convenient when multiplying radicals:
√2 ∗ √3 = √2 ∗ 3 = √6
Or,
3
3
3
3
√7 ∗ √49 = √7 ∗ 49 = √343 = 7
The index root has to be the same for the operation to take place. √2 ∗
3
√16 does not have this condition so the property cannot be applied
because one is a square root and the other a cubic root.
Then how can we simplify two roots with different index? Using the
3
radicals as exponents for the example √2 ∗ √16. We have
3
√2 ∗ √16 = 2
1⁄
2
∗ 16
1⁄
3
At plain sight it is not obvious but 16 is a power of 2:
2
1⁄
2
∗ 16
1⁄
3
= (21 )
1⁄
2
∗ (24 )
1⁄
3
Using the fifth law of exponents(𝑎𝑏 )𝑐 = 𝑎𝑏∗𝑐 it can be rearranged:
(21 )
1⁄
2
1⁄
3
=2
4⁄
3
=2
∗ (24 )
1⁄
2
∗2
4⁄
3
Now using 𝑎𝑏 ∗ 𝑎𝑐 = 𝑎𝑏+𝑐 , we get:
1⁄
2
2
∗2
11⁄
6
Therefore,
6
3
√2 ∗ √16 = √211
Combining 𝑎𝑏 ∗ 𝑎𝑐 = 𝑎𝑏+𝑐 ,
possible to say:
6
𝑐
𝑐
√𝑎 ∗ 𝑏 = √𝑎 ∗ √𝑏 and
𝑐
6
6
6
𝑏
𝑏
6
3
√2 ∗ √16 = √211 = √26 ∗ 25 = √26 ∗ √25 = 2 √25
𝟔
Therefore: √𝟐 ∗ √𝟏𝟔 = 𝟐 √𝟐𝟓 .
𝟑
𝑏
√𝑎𝑏 = ( √𝑎) = 𝑎, it is
The previous example was simplified because even if they had different
index roots, the radicand was the same by using prime factorization.
3
Another example that shows this process is √6 ∗ √12 simplified as follows:
Write the roots as exponents:
3
√6 ∗ √12 = 6
1⁄
2
∗ 12
1⁄
3
Find the prime factorization of the radicands (they are bases since they
are not in exponent form)
1⁄
2
=6
∗ 12
1⁄
3
= (2 ∗ 3)
1⁄
2
∗ (22 ∗ 3)
1⁄
3
Use the laws of exponents: (𝑎 ∗ 𝑏)𝑐 = 𝑎𝑐 ∗ 𝑏 𝑐 , 𝑎𝑏 ∗ 𝑎𝑐 = 𝑎𝑏+𝑐 , (𝑎𝑏 )𝑐 = 𝑎𝑏∗𝑐 and
the commutative property of multiplication:
= (2 ∗ 3)
= (2)
1⁄
2
1⁄
2
∗ (22 ∗ 3)
∗ (2)
2⁄
3
1⁄
3
∗ (3)
= (2)
1⁄
2
1⁄
2
∗ (3)
7⁄
6
=2
∗ (3)
1⁄
3
1⁄
2
=2
∗ (22 )
1⁄ +2⁄
2
3
1⁄
3
∗3
∗ (3)
1⁄
3
1⁄ +1⁄
2
3
5⁄
6
∗3
Reorganize:
=2
7⁄
6
𝑏
Use the law (𝑎𝑏 )𝑐 = 𝑎𝑏∗𝑐 , √𝑎 = 𝑎
= 27∗(
1⁄ )
6
1⁄ )
6
1
5⁄
6
= 27∗(
1⁄
𝑏
y √𝑎 ∗ 𝑏 = √𝑎 ∗ √𝑏
∗3
1
1⁄ )
6
𝑐
6
∗ 35∗(
𝑐
6
1⁄ )
6
𝑐
6
6
6
6
= (27 ) ⁄6 ∗ (35 ) ⁄6 = √27 ∗ √35 = √2 ∗ 26 ∗ √35 = 2 √2 ∗ √35
6
6
6
= 2 ∗ √2 ∗ 35 = 2√2 ∗ 243 = 2 √486
∗ 35∗(
Therefore, √6 ∗ √12 = 2√486
3
6
Now it is possible to have an idea on how different expressions can be
simplified even if the factors do not have the same root index. However,
it is easier to simplify with the same index.
The distributive property for radicals can also be used to simplify radical
expressions, as:
√18 = √2 ∗ 9 = √2 ∗ √9 = 3√2
2. Distributive property for dividing radicals.
It is expressed as:
𝒄
𝒄 𝒂
√𝒂 𝒄
√ = 𝒄 ; √𝒃 ≠ 𝟎
𝒃 √𝒃
It can be proven with the property:
𝑎 𝑐 𝑎𝑐
( ) = 𝑐 ; 𝑏𝑐 ≠ 0
𝑏
𝑏
In this way:
1
𝑎
𝑎 1⁄𝑐 𝑎 ⁄𝑐 √𝑎
√ =( ) = 1 =𝑐
𝑏
𝑏
𝑏 ⁄𝑐 √𝑏
𝑐
𝑐
It can be used to simplify expressions such as
3
√81
3
√3
:
3
3 81
√ = 3√27 = 3
=
3
3
√3
√81
Does it work the same when adding and subtracting radicals? This
question will be answered with two examples.
Simplify √2 + √2 − 4√2 + 6√2 using the notion of multiplication:
3
3
3
3
√2 + √2 − 4√2 + 6√2
3
= √2 + √2 − (√2 + √2 + √2 + √2)
3
3
3
3
3
3
+ ( √2 + √2 + √2 + √2 + √2 + √2)
Using the properties of addition:
3
3
3
3
3
3
3
3
3
3
3
3
3
= √2 + √2 − √2 − √2 − √2 − √2 + √2 + √2 + √2 + √2 + √2 + √2
3
= √2 − √2 − √2 − √2 − √2 + √2 + √2 + √2 + √2 + √2 + √2 + √2
Substitute: √2 = 𝑥 and √2 = 𝑧 to obtain:
3
=𝑥−𝑥−𝑥−𝑥−𝑥+𝑧+𝑧+𝑧+𝑧+𝑧+𝑧+𝑧
Combine like terms:
= −3𝑥 + 7𝑧
Substitute 𝑥 and 𝑧 with their original values:
3
3
3
√2 + √2 − 4√2 + 6√2 = −3√2 + 7√2
Although this process was very detailed, it is done to show several things,
including: the terms with radicals are real numbers, therefore, they
comply with all the properties of operations on the real set of numbers;
when renaming the radicals, it is evident that the key to simplifying
radicals, adding or subtracting them, is that they are similar, namely, that
the radicand and radical index of radicals are equal. Otherwise, the
operation cannot be performed as shown in the result.
3. Root of a root
The third property is expressed as:
𝒂 𝒃
√ √𝒄 =
𝒂∗𝒃
√𝒄
Proven by using the following law of exponents: (𝑎𝑏 )𝑐 = 𝑎𝑏∗𝑐 :
1⁄
𝑎
𝑎 𝑏
√ √𝑐 = ( 𝑐 1⁄𝑏 )
= 𝑐(
1⁄ )∗(1⁄ )
𝑎
𝑏
=𝑐
1⁄
𝑏∗𝑐
=
𝑎∗𝑏
√𝑐
It can be used to simplify expression such as √144:
12
12
√144 =
6
6∗2
6
2
√144 = √ √144 = √12
The following example combines properties of exponents and radicals to
simplify the algebraic expression:
𝑎𝑛 𝑏 𝑛+1 𝑐 𝑛+2
−3
(3
)
√2𝑎2 𝑏 3 𝑐 −4
3
3
1
3
3
[(2𝑎2 𝑏 3 𝑐 −4 ) ⁄3 ]
[ √2𝑎2 𝑏 3 𝑐 −4 ]
√2𝑎2 𝑏 3 𝑐 −4
= ( 𝑛 𝑛+1 𝑛+2 ) = 𝑛 𝑛+1 𝑛+2 3 = 3𝑛 3(𝑛+1) 3(𝑛+2)
(𝑎 𝑏 𝑐 )
𝑎 𝑏 𝑐
𝑎 𝑏
𝑐
3
3
(2𝑎2 𝑏 3 𝑐 −4 ) ⁄3
(2𝑎2 𝑏 3 𝑐 −4 )
= 3𝑛 3𝑛+3 3𝑛+6 = 3𝑛 3𝑛+3 3𝑛+6 = 2𝑎2−3𝑛 𝑏 3−(3𝑛+3) 𝑐 −4−(3𝑛+6)
𝑎 𝑏
𝑐
𝑎 𝑏
𝑐
2−3𝑛 3−3𝑛−3 −4−3𝑛−6
= 2𝑎
𝑏
𝑐
= 2𝑎2−3𝑛 𝑏 −3𝑛 𝑐 −3𝑛−10
2𝑎2−3𝑛
2−3𝑛 −3𝑛 −(3𝑛+10)
= 2𝑎
𝑏 𝑐
= 3𝑛 3𝑛+10
𝑏 𝑐
Did you know that…?
Karl Friedrich Gauss, one of the most important mathematicians in
history, proved the Fundamental Theorem of Algebra. This theorem has
𝑛
an interpretation on the topic being worked: √𝑎 has exactly n complex
roots.
Learning activity 2
1. In pairs, solve the following exercises showing your work step by
step. Remember to use the properties of exponents and radicals:
4
2
a. √25 ∗ √125
5
b.
√256
3
√1024
𝟑
𝟖
c. √𝟏𝟐𝟓
3
3
3
3
d. 3√6 + 2√16 − 4 √2 + 4√1296
e. ( 3
𝑥 8 𝑦 0 𝑧 −3
√27𝑥 2 𝑦 −4 𝑧
−6
)
2
f.
𝟓
𝟑𝟐𝒙𝟏𝟎 𝒚−𝟓 𝒛𝟏𝟓
√
𝟐
𝒙𝟐𝟓 𝒚𝟏𝟎𝒏 𝒛𝟓𝒃
(𝒙−𝟐)𝟖
g. √(𝒙−𝟐)𝟒
2. Share your results with your classmates and justify your process.
Consider this…
How can we find the value of n (exponent or number of factors)? Using
logarithms.
Activity 3
SKILL
Example 2:
Skill 1: Identify the rationalization of algebraic fractions.
Skill 2: Rationalize algebraic fractions with monomial denominators.
Skill 3: Rationalize algebraic fractions with binomial denominators.
𝑎
𝑛
√𝑥
is a fraction with a radical numerator. These fractions should be
rationalized, which is to eliminate that radical from the denominator. To
do so the fraction must not change. Two examples are used to explain the
process.
This process can be useful when simplifying algebraic expressions.
Example 1:
Rationalize the following expression
1
.
√2
Apply the following procedure:
1
∗
√2
√2 √2
√2
=
(√2)
2
=
√2
2
Remember that any number multiplied or divided by 1 is the same
number. Thus, the original expression does not change.
Rationalize the following expression
1
3
√32
1
3
. Apply the following procedure:
√3
3
√9
3
3
√9
√9
∗3
=3
=3
=
2
2
3
3
√3 √3
√3 ∗ 3
√3
3
3
Why are the examples multiplied by √32 and √2?; These numbers cancel
𝑏
𝑏
the radicals (the exponent becomes unitary) √𝑎𝑏 = ( √𝑎) = 𝑎.
This procedure can be generalized as follows:
𝑎
𝑛
√𝑥
𝑛
√𝑥 𝑛−1
∗𝑛
=
√𝑥 𝑛−1
𝑛
𝑎 √𝑥 𝑛−1
𝑛
√𝑥 𝑛
Which can be useful in cases such as:
𝑛
𝑎 √𝑥 𝑛−1
=
𝑥
𝑏
𝑥−1
4
√𝑥
The denominator is a monomial and can be rationalized as:
4
4
√𝑥 3
(𝑥 − 1) √𝑥 3
∗
=
4
4
𝑥
√𝑥
√𝑥 3
𝑥−1
Other expressions like
1
can be rationalized by using the notable
2+√3
2
2
product (𝑎 − 𝑏)(𝑎 + 𝑏) = 𝑎 − 𝑏 :
1
∗
2 − √3
2 + √3 2 − √3
=
2 − √3
(2)2 − (√3)
2
=
2 − √3
= 2 − √3
4−3
It also works with algebraic fractions with a binomial denominator as
shown next:
Example 1: rationalize the expression
𝑥+1
3 − √𝑥
∗[
3 + √𝑥
3 + √𝑥
]=
𝑥+1
3−√𝑥
(𝑥 + 1) ∗ (3 + √𝑥)
32 − √𝑥
2
=
(𝑥 + 1) ∗ (3 + √𝑥)
9−𝑥
Use the notable product (𝑎 − 𝑏)(𝑎 + 𝑏) = 𝑎2 − 𝑏 2 ; also, 3 + √𝑥 is the
conjugate of 3 − √𝑥. In general, the conjugate of an expression 𝑎 + 𝑏√𝑥 is
𝑎 − 𝑏√𝑥. Thus, to rationalize algebraic fractions with this type of
denominators, multiply the numerator and denominator by the conjugate
of the denominator.
Example 2: Rationalize the expression
3−𝑎
3
2
3
( √𝑎 ) − 𝑏 √𝑎 − 𝑏 2
.[
]=
2
√𝑎 + 𝑏 ( 3√𝑎) − 𝑏 3√𝑎 − 𝑏 2
3
3−𝑎
3
√𝑎+𝑏
2
(3 − 𝑎) [( 3√𝑎) − 𝑏 3√𝑎 − 𝑏 2 ]
𝑎 + 𝑏3
Use the notable product (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏 2 ) = 𝑎3 − 𝑏 3
The general way to rationalize algebraic fractions is:
𝑝(𝑥)
𝑛
√𝑎 ± 𝑏
∗[
𝑛
𝑛−1
𝑛
𝑛−1
( √𝑎 )
( √𝑎 )
=
𝑛
𝑛−2
𝑛
𝑛−2
𝑛
𝑛−1
± ( √𝑎 )
± ( √𝑎 )
𝑝(𝑥) [( √𝑎)
𝑛
𝑛−3 2
𝑛
𝑛
𝑛−3
𝑛
𝑏 ± ( √𝑎 )
𝑏 ± ( √𝑎 )
𝑛
± ( √𝑎 )
𝑏 ± ⋯ ± ( √𝑎)𝑏 𝑛−2 ± 𝑏 𝑛−1
𝑛−2
𝑏 2 ± ⋯ ± ( √𝑎)𝑏 𝑛−2 ± 𝑏 𝑛−1
𝑛
𝑏 ± ( √𝑎 )
𝑎 ± 𝑏𝑛
Learning activity 3
1. Rationalize the following expressions:
2
a.
3
b.
5
√2
6
√16
𝑛−3 2
𝑛
]
𝑏 ± ⋯ ± ( √𝑎)𝑏 𝑛−2 ± 𝑏 𝑛−1 ]
c.
d.
𝑥 2 −2
3
√𝑥𝑦
2𝑎−𝑏
2
√𝑎−𝑏
2. Share and justify to your classmates.
Abstract
Exponents and radicals are closely related. Such relationship can be
evidenced in the definition:
𝑏
√𝑐 = 𝑎 ↔ 𝑎 𝑏 = 𝑐
And in the expressions:
𝑏
𝑏
𝑏
√𝑎𝑏 = ( √𝑎) = 𝑎.
𝒃
𝟏⁄
𝒃
√𝒂 = 𝒂
The exponent laws are:
1. 𝒂𝒃 ∗ 𝒂𝒄 = 𝒂𝒃+𝒄
2.
𝒂𝒃
𝒂𝒄
= 𝒂𝒃−𝒄 ; 𝒂𝒄 ≠ 𝟎
3. (𝒂 ∗ 𝒃)𝒄 = 𝒂𝒄 ∗ 𝒃𝒄
𝒂 𝒄
𝒂𝒄
4. (𝒃) = 𝒃𝒄 ; 𝒃𝒄 ≠ 𝟎
𝒄
5. (𝒂𝒃 ) = 𝒂𝒃.𝒄
𝟏
6. 𝒂−𝒃 = 𝒂𝒃 ; 𝒂𝒃 ≠ 𝟎
𝒂 −𝒄
7. (𝒃)
𝒃 𝒄
= (𝒂)
The radical properties are:
1. √𝒂. 𝒃 = √𝒂. √𝒃
𝒄
𝒄
𝒂
2. √𝒃 =
𝒂
𝒄
𝒄
𝒄
√𝒂 𝒄
; √𝒃
√𝒃
𝒄
𝒃
3. √ √𝒄 =
≠𝟎
𝒂∗𝒃
√𝒄
Remember that…
-
These properties serve to simplify arithmetic and algebraic
expressions.
There are no properties for addition and subtraction of numbers or
algebraic terms expressed as powers or roots.
To add radicals, these must have the same radical index and the
same radicand, as if they were similar terms.
-
Rationalization is a process by which a radical expression is
removed from the denominator of a fraction.
To add radicals, these must have the same radical index and the
same radicand, as if they were similar terms.
Rationalization is a process by which a radical expression is
removed from the denominator of a fraction.
Depending on whether the denominator is a monomial or a
binomial, rationalization is generalized as follows:
𝑛
√𝑥 𝑛−1
𝑎
𝑛
√𝑥
𝑛
∗𝑛
=
√𝑥 𝑛−1
𝑎 √𝑥 𝑛−1
𝑛
√𝑥 𝑛
𝑛
𝑎 √𝑥 𝑛−1
=
𝑥
or
𝑝(𝑥)
𝑛
√𝑎 ± 𝑏
∗[
𝑛
𝑛−1
𝑛
𝑛−1
( √𝑎 )
( √𝑎 )
=
𝑛
𝑛−2
𝑛
𝑛−2
𝑛
𝑛−1
± ( √𝑎 )
± ( √𝑎 )
𝑝(𝑥) [( √𝑎)
𝑛
𝑛−3 2
𝑛
𝑛
𝑛−3
𝑛
𝑏 ± ( √𝑎 )
𝑏 ± ( √𝑎 )
𝑛
± ( √𝑎 )
𝑏 ± ⋯ ± ( √𝑎)𝑏 𝑛−2 ± 𝑏 𝑛−1
𝑛−2
𝑏 2 ± ⋯ ± ( √𝑎)𝑏 𝑛−2 ± 𝑏 𝑛−1
𝑛
𝑏 ± ( √𝑎 )
𝑛−3 2
]
𝑛
𝑏 ± ⋯ ± ( √𝑎)𝑏 𝑛−2 ± 𝑏 𝑛−1 ]
𝑎 ± 𝑏𝑛
respectively.
Homework
1. Use the exponent or radical properties to simplify the following
expressions:
2
4
a. √32 ∗ √512
𝟑
𝟐𝟏𝟔
b. √𝟏𝟐𝟓
3
3
4
4
c. 3√4 + 2√4 − 4√4 + 12√4
d. ( 3
2𝑥 3 𝑦 4 𝑧 2
√1453𝑥 2 𝑦−4 𝑧
𝟔
𝟔𝟒𝒙𝟏𝟐 𝒚−𝟔 𝒛𝟏𝟖
𝟒
(𝟑𝒙+𝟐𝒚)𝟏𝟐
e. √ 𝒙𝟐𝟒 𝒚𝟏𝟐𝒏 𝒛𝟔𝒃
f.
√ (𝟑𝒙+𝟐𝒚)𝟒
2. Rationalize:
4
a.
3
b.
5
√3
12
√5
0
)
2
c.
d.
e.
2𝑥 2
3
√𝑥+𝑦
2𝑎+𝑏
2
√𝑎𝑏
𝑎𝑏
2
√𝑎+2
3. Share and justify the results obtained in your homework.
Evaluation
After studying the concepts and methodologies to understand the
exponents and radicals in real numbers, an evaluation will be performed
to allow you visualize what you learned and improve the proposed skills.
These concepts will help you in many practical situations of real and
academic life. It is important to analyze each proposed item carefully
and indicate the correct answer according to the topic.
Select the correct answer in the following questions related to
exponents, radicals, their corresponding properties, and rationalization:
Skills:
- Find positive exponents of real numbers.
1. The square root of the 121 is:
a.
b.
c.
d.
11
−11
±11
10,59
Answer key:
Correct answer: C
Because (-11)(-11)=121=(11)(11)
Incorrect answer: Review the definition of radicals and the case of roots
with odd index.
Skills:
-
Deduct the properties of radicals based on the relation with
exponents
2. One of the following expressions is NOT a property of radicals:
𝑛
a. √𝑎 + 𝑏 = √𝑎 + √𝑏.
𝒄
𝒄
𝒄
b. √𝒂. 𝒃 = √𝒂. √𝒃
𝑛
𝒄
𝑛
𝒂
c. √𝒃 =
𝒂
𝒄
√𝒂 𝒄
; √𝒃
√𝒃
𝒄
𝒃
d. √ √𝒄 =
≠𝟎
𝒂∗𝒃
√𝒄
Answer key:
Correct answer: a
It is not a property of radicals because, for example, √1 + 4 ≠ √1 + √4
2
2
2
Incorrect answer: Review the properties of the radicals.
Skills:
Deduct the properties of exponents through generalizations of cases
and patterns.
3. Andres and his friend Carlos discuss about a point of their math
homework, since Andres claims that one of the properties of exponents is
𝑎𝑏 ∗ 𝑐 𝑑 = (𝑎 ∗ 𝑐)𝑑+𝑏 , and Carlos considers that this is not true. Who is right?
-
a. Andres because 11 ∗ 41 = (1 ∗ 4)1+1
b. Carlos because 23 ∗ 45 ≠ (2 ∗ 4)3+5
c. Carlos because 23 ∗ 45 = (2 ∗ 4)3+5
d. Andres because 11 ∗ 41 ≠ (1 ∗ 4)1+1
Answer Key:
Correct answer: b. It is not a property of exponents because, for
example, 23 ∗ 45 ≠ (2 ∗ 4)3+5
Incorrect answer: Review the properties of exponents.
Skills:
Identify the rationalization of algebraic fractions.
Rationalize algebraic fractions with monomial denominators.
4. Based on the rationalization of fractions with monomial denominators,
5
the rationalization of the expression 2√3 is:
-
a.
b.
c.
d.
5√3
3
5√3
6
√3
3
√3
6
Answer key:
5
√3
Correct answer: 2√3 ( ) =
√3
5√3
2(√3)
2
=
5√3
6
Incorrect answer: Review the procedure to rationalize.
Skills:
- Apply the properties of exponents to simplify algebraic expressions.
5. Using the properties of exponents, what is the value of the expression
[((𝑥𝑦)𝑛 )0 ]𝑘 ?
a. 1
b. 0
c. 𝑥𝑦
d. (𝑥𝑦)𝑛𝑘
Answer key:
Correct answer: a. since 𝑎0 = 1
Incorrect answer: Review the properties and definition of exponents.
Bibliography
Stewart, J. (2007). (5ta edición). Precálculo, Matemáticas para el cálculo.
Cengage Learning.
Stewart, J. (2012). (6ta edición). Precálculo, Matemáticas para el cálculo.
Cengage Learning.
Glossary
Exponents: if 𝑎 is a real number and 𝑛 is a positive number, then
𝒏𝒕𝒉 𝒑𝒐𝒘𝒆𝒓 of 𝑎 is
𝑎𝑛 = ⏟
𝑎. 𝑎. 𝑎. 𝑎. … . 𝑎
𝑛 𝑓𝑎𝑐𝑡𝑜𝑟𝑠
𝑎 is the base, and 𝑛 is the exponent.
Radicals: if 𝑛 is any positive number, then the 𝒏𝒕𝒉 𝒓𝒐𝒐𝒕 of a is defined
as:
√𝑎 = 𝑏 means 𝑏 𝑛 = 𝑎
𝑛
If 𝑛 is even, then 𝑎 ≥ 0 and 𝑏 ≥ 0.
Definitions retrieved from:
Stewart, J. (2012). (6ta edición). Precálculo, Matemáticas para el cálculo.
Cengage Learning.
Vocabulary box
Once upon a time: a long time ago.
Subjects: people that work for a sultan; commonly slaves
Cheer him up: To improve someone’s mood
Array: large group of things
Undermined: to give little or no importance
Delay: to extend time, to postpone
English Review Topic
Present Tense, Adjectives.