My name is Erna Apriliana, usually called Erna and number ID 07305141011. I want to tell about my experience with my friend. My friend is Zulvina Tri Susanti. I had explained about Heating and Cooling of Building.
I choose topic “Heating and Cooling of Building”, because I was inspirited by study about Linear Equation in Differential Equation. From that topic, we can know to determine the building temperature.
Three main factors affecting the temperature inside the building. The first factor is the heat produced by people, lights, and machines inside the building. This causes a rate of increase in temperature that we will denote by H(t). The second factor is the heating or cooling supplied by the furnace (or air conditioning). This rate of increase or decrease in temperature we will by U(t).
In general, the additional heating rate H(t) and the furnace (or air conditioning) rate U(t) are described in terms of energy per unit time. The third factor is the effect of the outside temperature M(t) on the temperature inside the building. This is known as Newton’s Law of cooling, which states that there is a rate of change in temperature T(t) that is proportional to the difference between the outside temperature M(t) and the inside temperature T(t). That is, the rate of change in the building temperature due to M(t) is :
dT(t) / dt = K [ M(t) – T(t) ]
the positive constant K depends on the physical properties of the building, but K does not depends on M, T, or t. When the outside temperature is grater than the inside temperature, then M(t) - H(t) > 0, and there is an increase in the rate of change of the building temperature due to M(t). When the outside temperature is less than the inside temperature, then M(t) - H(t) < 0, and there is an decrease in the rate of change.
Summarizing, we find
(1) dT(t) / dt = K [ M(t) – T(t) ] + H(t) +U(t)
Where the additional heating rate H(t)
is always non negative and U(t is positive for nace heating and negative for air conditioner cooling.
Equation (1) is linear, it can be solved using the method with standards form
(2) dT(t) / dt + P(t)T(t) = Q(t)
where P(t) = K
(3) Q(t) = KM(t + H(t) + U(t)
We find that the integrating factor is
Μ(t) = exp ( ∫ K dt ) = e^Kt
To solve (2), multiply each side by e^Kt and integrate:
e^Kt dT(t) / dt + K e^Kt P(t)T(t) = e^Kt Q(t)
e^Kt T(t) = ∫ e^Kt Q(t) dt + C
solving for T(t) gives
(4) T(t) = e^-Kt ∫ e^Kt Q(t) dt + C e^-Kt
= e^-Kt {∫ e^Kt [ KM(t) + H(t) + U(t) ] dt + C}
In this process, I don’t have any troubles to explain that. Santi could understand what I had explained. To make sure that she has understood, I gave her question like this :
For example, at the end of the day (at time t0), when people leave the building, the outside temperature stays constant at M0, the additional heating rate H inside the building is zero, and the air conditioner rate U is zero. Determine T(t), the initial condition T(t0) = T0
And her result is :
M = M0, H = 0, U = 0
From equqtion (4) she found :
T(t) = e^-Kt {∫ e^Kt [ KM(t) + H(t) + U(t) ] dt + C}
= e^-Kt {∫ e^Kt [ KM(t) + 0+ 0 ] dt + C}
= e^-Kt {∫ e^Kt [ KM(t) ] dt + C}
= e^-Kt [ M0 e^Kt + C ]
= M0 + C e^-Kt
That was my experience of studying English 2 with the lecturer is Mr. Marsigit, MA. I’m feel so exciting.
Kamis, 15 Januari 2009
video
VIDEO 1
A. GRAMMAR
Jenis / tipe kalimat yang paling dasar adalah kalimat sederhana . Karena semua elemen atau bagian didalam kalimat merupakan bagian dari subyek atau predikat.
Subyek kalimat adalah bagian yang melakukan atau mengerjakan tindakan dari kata kerja utama.
Subyek sederhana adalah kata benda tertentu ( spesifik ) yang menampilkan atau melakukan tindakan.
Sebagai contoh :
Anak kecil yang bahagia menendang boneka kerdil melewati pagar.
Anak adalah kata benda yang melakulan tindakan yaiti mrnrndang.
Anak adalah subyek sederhana.
Karena kecil, bahagia melekat pada anak maka,
Anak kecil yang bahagia disebut subyek komplit.
“Kalimat sederhana”
Predikat suatu kalimat terdiri atas :
Kata kerja utama + Segala sesuatu yang berkaitan dengannya.
(predikat sederhana)
Segala sesuatu tadi yang bukan merupakan bagian dari subyek.
Keseluruhan dari kata kerja utama + Segala sesuatu yang berkaitan dengannya tersebut disebut Predikat komplit.
- Predikat
Anak kecil yamg bahagia menedang boneka kerdil melewati pagar.
* Menendang adalah predikat sederhana
* Boneka kerdil adalah apa yang ditendang
* Melewati pagar adalah menerangkan dmana menendang.
Ketiganya diatas merupakan predikat komplit.
Jadi dalam kalimat :
Anak kecil yang bahagia :Subyek komplit/lengkap
Menendang boneka kerdil melewati pagar : Predikat komplit / lengkap.
Kadang-kadang kalimat sederhana dapat diperoleh tanpa sebuah subyek atau predikat.
Kalimat peringkat adalah kalimat yang ditujukan terhadap orang kedua yaitu “kamu”.
Kata “kamu” keluar dari kalimat dan hanya tersamar kalimat perintah ( contoh ) :
Tendang boneka kerdil iu melewati pagar.
* Tendang :Predikat sederhana
* Tendang boneka kerdil iu melewati pagar :Predikat lengkap
Subyeknya dimana? Tidak ada
Jika aku memerintah kamu, tendang boneka kerdil itu melewati pagar.
Siapa yang harus melakukannya ? Ya, kamu.
Sebenarnya kalimar begini. “Hei, kamu, tendang boneka kerdil itu melewati pagar.”
Subyek “kamu” tersamar.
Siapa yang menendang boneka kerdil itu melewati pagar ? Cindy * Cindy :Predikat yang tersamar
Cindy hanyalah satu kata.
Dapat dikatakan dalam kalimat lain adalah :
“Cindy,tendang boneka kerdil itu melewati pagar.”
B. Frase dan Obyek
Video 2 ( kata kerja )
Apakah kata kerja itu?
Kata kerja adalah suatu kata yang menunjukkan tindakan atau mendeskripsikan keadaan sesuatu apa yang kata benda dan kata ganti sedang lakukan.
Kata kerja merupakan salah satu bagian terpenting dalam kalimat.
Contoh kalimat pendek :
Dave berlari
Kata kerja, karena menunjukkan apa yang dilakukan Dave
Apa yang Dave lakukan? Ia berlari.
Dalam Grammar Bahasa Inggris, kata kerja mengubah bentuk untuk menunjukkan siapa yang melakukan tindakan.
Siapa yang melakukan tindakan?
I do, You do, he does, we do, they do.
• I run
• You run
• He run
• She run
• It run
• We run
• They run
Apa yang barusan diatas adalah mengambil sebuah kata kerja dan melihat bagaimana itu berubah dengan subyek yang berbeda. Itu adalah hubungan
Kata kerja –to be
Kita mengatakan :
• I am
• You are
• He is
• She is
• It is
• We are
• They are
I,you,he,she,it = kata ganti dan merupakan subyek dan merupakan subyek tunggal. Hanya ada satu kata benda.
We,they = subyek jamak kata benda yang mempunyai lebih dari satu anggota.
Kata benda tunggal memakai kata kerja tunggal.
Kata kerja jamak memakai kata kerja jamak.
Mrs. Midori Yodels.
Yodels adalah bentuk dari yodel yang diguna.
Saudara- saudara Midori = else, gretel, hera
Saudara- saudara Midori = kata-kata jamak, sehingga menggunakan kata kerja jamak.
The Midori sisters Yodel
Yodel = bentuk jamak dari Yodel
Petunjuk Grammar :
Kata benda tunggal memakai kata kerja tunggal.
Kata benda jamak mengubah bentuk tergantung pada point dalam waktu apa tindakan itu terjadi.
Kita menyebut perubahan ini dengan sebutan “Tense”
Video 3 ( Kata Keterangan)
Kata keterangan adalah kata yang menjelaskan atau menerangkan kata kerja dan juga bisa menerangkan kata sifat, dan kata keterangan yang lain.
Jika kita menjawab pertanyaan :
• Bagaimana?
• Sebagaimana sering?
• Kapan?
• Atau Untuk apa meluas?
Kamu dapat menggunakan kata keterangan
Kata Sifat + Ly
Contoh :
Sebagai pembicara, Simon seharusnya pelan, tapi dia berbicara dengan lambat.
(slowly)
Slow menjelaskan Simon “ slow ” = kata sifat
Slowly menjawab “Bagaimana Simon berbicara? ”
Slowly = kata keterangan
Slow + ly = kata keterangan
Contoh:
The color of this mushrooms is slightly different.
Slight + Ly =kata keterangan untuk menjelaskan kata different.
Contoh :
This mushrooms is very definitely poisonous.
very : Kt.ket, definitely : Kt.ket, poisonous :Kt.sifat
Very adalah kata keterangan yang menjelaskan kata keterangan lain.
Very tanpa akhiran Ly, suatu pengecualian.
Good atau Well mana kata keterangan?
Menjawab bagaimana?
Menjwab dengan kata keterangan = menggunakan “ well ”
Candace can play the accordion very well.
well : Kata keterangan dari bentuk good
Candace’s playing is good.
Good menjelaskan kata kata playing.
Playing disini adalah Gerund.
Gerund adalah kata kerja dalam bentuk –ing digunakan sebagai kata benda dalam kalimat.
Karena yang diterangkan kata benda, kita menggunakan kata sifat “ Good ”
Video 4 ( Trigonometri )
It is from ancient treet and can means triangle and meter. All Trigonometri is related to study a right triangle & relationship between the sides and the angle of the right triangle.
Let’s start with a right triangle.
With the sides of 3, 4, 5 and the hypotenuse hypotenuse : 5
adjacent:3
opposite : 4
Want to define Sin θ ?
You can see
Son can toa
Sin is opposite over hypotenuse
Cos is adjacent over hypotenuse
Tan is opposite over adjacent
Sin∂ = Opp / Hyp
The opposite side of θ I 4
The hypotenuse is the longest side of triangle ( right triangle)
So,
Sin θ ………?
Sin θ = opp / hyp
= 4 / 5
By the reasoning the figure, you can see that the adjacent side is 3 (because 4 is opposite side of θ ) and 5 is the hypotenuse.
Cos θ …….?
Cos θ = adj / hyp
= 3 / 5
What is the Tan θ…….?
Tan θ = opp / adj
= 3 / 4
So, The Tan θ is equal 4 / 3
Video 5 Compound sentences ( Kalimat Majemuk )
It’s the end of the world as we know it and I feel fine
* Ada 2 klausa yang dihubungkan dengan 1 kata penghubung ( and )
* Ketika suatu kalimat digunakan sebagai bagian-bagian dalam kalimat yang lebih besar, maka kalimat yang lebih kecil tersebut disebut klausa.
* Ketika sebuah klausa dapat berdiri sendiri dalam sebuah kalimat maka, klausa tersebut disebut “ Independent clause ”
* Jika kita memiliki 2 klausa dalam 1 kalimat maka, kalimat itu disebut majemuk.
* Untuk menggabungkan 2 Independent clause,
Kita menggunakan =
* Titik dua ( : ) , ketika klausa ke-2 menjelaskan klausa ke-1,
“ I love my two sister’s, they bake me pie”
Untuk menggabungkan kalimat menjadi kalimat majemuk kita gunakan titik dua ( : )
* Titik koma ( ; ) , untuk menggantikan konjungsi.
“It’s the end of the world and I feel fine. ”
A. GRAMMAR
Jenis / tipe kalimat yang paling dasar adalah kalimat sederhana . Karena semua elemen atau bagian didalam kalimat merupakan bagian dari subyek atau predikat.
Subyek kalimat adalah bagian yang melakukan atau mengerjakan tindakan dari kata kerja utama.
Subyek sederhana adalah kata benda tertentu ( spesifik ) yang menampilkan atau melakukan tindakan.
Sebagai contoh :
Anak kecil yang bahagia menendang boneka kerdil melewati pagar.
Anak adalah kata benda yang melakulan tindakan yaiti mrnrndang.
Anak adalah subyek sederhana.
Karena kecil, bahagia melekat pada anak maka,
Anak kecil yang bahagia disebut subyek komplit.
“Kalimat sederhana”
Predikat suatu kalimat terdiri atas :
Kata kerja utama + Segala sesuatu yang berkaitan dengannya.
(predikat sederhana)
Segala sesuatu tadi yang bukan merupakan bagian dari subyek.
Keseluruhan dari kata kerja utama + Segala sesuatu yang berkaitan dengannya tersebut disebut Predikat komplit.
- Predikat
Anak kecil yamg bahagia menedang boneka kerdil melewati pagar.
* Menendang adalah predikat sederhana
* Boneka kerdil adalah apa yang ditendang
* Melewati pagar adalah menerangkan dmana menendang.
Ketiganya diatas merupakan predikat komplit.
Jadi dalam kalimat :
Anak kecil yang bahagia :Subyek komplit/lengkap
Menendang boneka kerdil melewati pagar : Predikat komplit / lengkap.
Kadang-kadang kalimat sederhana dapat diperoleh tanpa sebuah subyek atau predikat.
Kalimat peringkat adalah kalimat yang ditujukan terhadap orang kedua yaitu “kamu”.
Kata “kamu” keluar dari kalimat dan hanya tersamar kalimat perintah ( contoh ) :
Tendang boneka kerdil iu melewati pagar.
* Tendang :Predikat sederhana
* Tendang boneka kerdil iu melewati pagar :Predikat lengkap
Subyeknya dimana? Tidak ada
Jika aku memerintah kamu, tendang boneka kerdil itu melewati pagar.
Siapa yang harus melakukannya ? Ya, kamu.
Sebenarnya kalimar begini. “Hei, kamu, tendang boneka kerdil itu melewati pagar.”
Subyek “kamu” tersamar.
Siapa yang menendang boneka kerdil itu melewati pagar ? Cindy * Cindy :Predikat yang tersamar
Cindy hanyalah satu kata.
Dapat dikatakan dalam kalimat lain adalah :
“Cindy,tendang boneka kerdil itu melewati pagar.”
B. Frase dan Obyek
Video 2 ( kata kerja )
Apakah kata kerja itu?
Kata kerja adalah suatu kata yang menunjukkan tindakan atau mendeskripsikan keadaan sesuatu apa yang kata benda dan kata ganti sedang lakukan.
Kata kerja merupakan salah satu bagian terpenting dalam kalimat.
Contoh kalimat pendek :
Dave berlari
Kata kerja, karena menunjukkan apa yang dilakukan Dave
Apa yang Dave lakukan? Ia berlari.
Dalam Grammar Bahasa Inggris, kata kerja mengubah bentuk untuk menunjukkan siapa yang melakukan tindakan.
Siapa yang melakukan tindakan?
I do, You do, he does, we do, they do.
• I run
• You run
• He run
• She run
• It run
• We run
• They run
Apa yang barusan diatas adalah mengambil sebuah kata kerja dan melihat bagaimana itu berubah dengan subyek yang berbeda. Itu adalah hubungan
Kata kerja –to be
Kita mengatakan :
• I am
• You are
• He is
• She is
• It is
• We are
• They are
I,you,he,she,it = kata ganti dan merupakan subyek dan merupakan subyek tunggal. Hanya ada satu kata benda.
We,they = subyek jamak kata benda yang mempunyai lebih dari satu anggota.
Kata benda tunggal memakai kata kerja tunggal.
Kata kerja jamak memakai kata kerja jamak.
Mrs. Midori Yodels.
Yodels adalah bentuk dari yodel yang diguna.
Saudara- saudara Midori = else, gretel, hera
Saudara- saudara Midori = kata-kata jamak, sehingga menggunakan kata kerja jamak.
The Midori sisters Yodel
Yodel = bentuk jamak dari Yodel
Petunjuk Grammar :
Kata benda tunggal memakai kata kerja tunggal.
Kata benda jamak mengubah bentuk tergantung pada point dalam waktu apa tindakan itu terjadi.
Kita menyebut perubahan ini dengan sebutan “Tense”
Video 3 ( Kata Keterangan)
Kata keterangan adalah kata yang menjelaskan atau menerangkan kata kerja dan juga bisa menerangkan kata sifat, dan kata keterangan yang lain.
Jika kita menjawab pertanyaan :
• Bagaimana?
• Sebagaimana sering?
• Kapan?
• Atau Untuk apa meluas?
Kamu dapat menggunakan kata keterangan
Kata Sifat + Ly
Contoh :
Sebagai pembicara, Simon seharusnya pelan, tapi dia berbicara dengan lambat.
(slowly)
Slow menjelaskan Simon “ slow ” = kata sifat
Slowly menjawab “Bagaimana Simon berbicara? ”
Slowly = kata keterangan
Slow + ly = kata keterangan
Contoh:
The color of this mushrooms is slightly different.
Slight + Ly =kata keterangan untuk menjelaskan kata different.
Contoh :
This mushrooms is very definitely poisonous.
very : Kt.ket, definitely : Kt.ket, poisonous :Kt.sifat
Very adalah kata keterangan yang menjelaskan kata keterangan lain.
Very tanpa akhiran Ly, suatu pengecualian.
Good atau Well mana kata keterangan?
Menjawab bagaimana?
Menjwab dengan kata keterangan = menggunakan “ well ”
Candace can play the accordion very well.
well : Kata keterangan dari bentuk good
Candace’s playing is good.
Good menjelaskan kata kata playing.
Playing disini adalah Gerund.
Gerund adalah kata kerja dalam bentuk –ing digunakan sebagai kata benda dalam kalimat.
Karena yang diterangkan kata benda, kita menggunakan kata sifat “ Good ”
Video 4 ( Trigonometri )
It is from ancient treet and can means triangle and meter. All Trigonometri is related to study a right triangle & relationship between the sides and the angle of the right triangle.
Let’s start with a right triangle.
With the sides of 3, 4, 5 and the hypotenuse hypotenuse : 5
adjacent:3
opposite : 4
Want to define Sin θ ?
You can see
Son can toa
Sin is opposite over hypotenuse
Cos is adjacent over hypotenuse
Tan is opposite over adjacent
Sin∂ = Opp / Hyp
The opposite side of θ I 4
The hypotenuse is the longest side of triangle ( right triangle)
So,
Sin θ ………?
Sin θ = opp / hyp
= 4 / 5
By the reasoning the figure, you can see that the adjacent side is 3 (because 4 is opposite side of θ ) and 5 is the hypotenuse.
Cos θ …….?
Cos θ = adj / hyp
= 3 / 5
What is the Tan θ…….?
Tan θ = opp / adj
= 3 / 4
So, The Tan θ is equal 4 / 3
Video 5 Compound sentences ( Kalimat Majemuk )
It’s the end of the world as we know it and I feel fine
* Ada 2 klausa yang dihubungkan dengan 1 kata penghubung ( and )
* Ketika suatu kalimat digunakan sebagai bagian-bagian dalam kalimat yang lebih besar, maka kalimat yang lebih kecil tersebut disebut klausa.
* Ketika sebuah klausa dapat berdiri sendiri dalam sebuah kalimat maka, klausa tersebut disebut “ Independent clause ”
* Jika kita memiliki 2 klausa dalam 1 kalimat maka, kalimat itu disebut majemuk.
* Untuk menggabungkan 2 Independent clause,
Kita menggunakan =
* Titik dua ( : ) , ketika klausa ke-2 menjelaskan klausa ke-1,
“ I love my two sister’s, they bake me pie”
Untuk menggabungkan kalimat menjadi kalimat majemuk kita gunakan titik dua ( : )
* Titik koma ( ; ) , untuk menggantikan konjungsi.
“It’s the end of the world and I feel fine. ”
Jumat, 09 Januari 2009
definition of rectangle
Rectangle
This time, I would to tell you about “Rectangle”
The definition of a rectangle is quadrilateral that has four right angle, in other word the value of each angle is 90 degree.
For example : rectangle of ABCD, there are four sides : AB-BC-CD-AD
Side AB and side DC is parallel, side AC and side BD is parallel.
Side AD adnd side BC are diagonal a rectangle. Side AD is 12 cm long, side CD is 8 cm long. Determine long of side AC !
Solution :
We will determine long of side AC.
To find long of side AC we use the Pythagoras theorem because angle of ACD is a right angle.
AD^2 = AC^2 + CD^2
12^2 = AC^2 + 8^2
AC^2 = 144 – 64
= 80
AC = root of 80
= 8,94
So, long of side AC is 8,94 cm.
That’s about rectangle.
I think is enough. Thank you for your attention.
This time, I would to tell you about “Rectangle”
The definition of a rectangle is quadrilateral that has four right angle, in other word the value of each angle is 90 degree.
For example : rectangle of ABCD, there are four sides : AB-BC-CD-AD
Side AB and side DC is parallel, side AC and side BD is parallel.
Side AD adnd side BC are diagonal a rectangle. Side AD is 12 cm long, side CD is 8 cm long. Determine long of side AC !
Solution :
We will determine long of side AC.
To find long of side AC we use the Pythagoras theorem because angle of ACD is a right angle.
AD^2 = AC^2 + CD^2
12^2 = AC^2 + 8^2
AC^2 = 144 – 64
= 80
AC = root of 80
= 8,94
So, long of side AC is 8,94 cm.
That’s about rectangle.
I think is enough. Thank you for your attention.
"Rational Exponen, Form Of Root, and Logarithm"
Chapter 1
"Rational Exponen, Form Of Root, and Logarithm"
Realization of base Competence which is showed with the study result bellow.
1. Understanding and using characteristic and rule of exponen, root, and logarithm in solving problem
2. Doing algebra manipulation in technical which is related to exponen, root, and logarithm
Rainbow is nature indication which is seen at the pouring rain. When sun in our back, length of wave and frequency of rainbow color are different. Try to find out and connected with material in this chapter!
You have heard about exponential number, right?Do you know, many cases in our life which is notatined in exponential number. For example length of wave and frequency of rainbow's color. Purple wave length of rainbow is 3,9 x 10^-7 metre until 7,7 x 10^14 metre and the frequency 6,9 x 10^14 Hertz until 7,7 x 10^14 Hertz. Another example, the length of DNA string (deoxcyribonuleic acid) in a cell 10^-7 metre and average humans body consist of 10^4 cells. DNA is part of main cell which involve in protein synthetic and genetic factor. The number of 10^-7 and 10^14 are exponen numbers. Try to find out the other examples.
A. Exponential of Positive Integer
In addition there is a process sum of n-th times can be written as
3 + 3 + 3 + 3 + 3 = 5 x 3
in other case of multiplication, there is a multiplication of n-th time can be written as
3 x 3 x 3 x 3 x 3 = 3^5
3^5 is called exponential number, 3 is called main of number and 5 is called exponen.
3^5 is called three to the power of five
in general form :
if a is real number and n is positive integer number, so :
a^n = a x a x a x …x a → n times factor
a^n read “ a exponen n “ with a is called main of number and n is called exponen.
Characteristics of exponential number with positive integer exponen.
Characteristics 1 (Multiplication of Exponential Number) :
If two exponential number or more that have same main of number, so the exponen must be sum.
Example 1.1 :
Determine multiplication result bellow :
a. 5^3 x 5^4
b. 5 x 5^3
c. 13^2 x 13^5 x 13
Answer :
a. 5^3 x 5^4 = 5 (3+4) = 5 7
b. 5 x 5^3 = 5(1+3) 5 4
c. 13^2 x 13^5 x 13 = 13(2+5+1) = 138
In general form :
If a is real number and m, n are positive integer, so :
a^m x a^n = a(m+n)
Characteristics 2 (Division of Exponential Number)
If an exponential number is divided to the other exponen number that have same main of number so the exponen have to be subtracted.
Example 1.2 :
Determine division result bellow :
1. 2^3 : 2
2. 5^6 : 5^4
3. 8^7 : 8^7
Answer :
1. 2^3 : 2 = 2 (3-1)
= 2^2
= 4
2. 5^6 : 5^4 = 5(6-4)
= 5^2
= 25
3. 8^7 : 8^2 = 8(7-2)
= 8^2
In general form :
If a is real number and m,n are positive integer, so :
a^m – a^n = a(m-n)
with a ≠ 0, m > 0
Characteristics 3 (Exponen of Exponential Number)
If an exponential number exponentized to the other number, so the exponen must be multiplicated.
Example 1.3 :
Determine exponen result bellow :
1. (3^3)^2
2. (5^2)^4
3. (10^3)^7
Answer :
1. (3^3)^2 = 3^3x2
= 3^6
2. (5^2)^4 = 5^2x4
= 5^8
3. (10^3)^7 = 10^3x7
= 10^21
In general form :
If a is real number and m,n are positive integer, so :
(a^m)^n = a^mxn
Characteristics 4 (Exponen of Multiplication Number)
If the multiplication of two number or more be exponented, so eachs number must be exponented,
Example 1.4 ;
Determine exponent result bellow ;
a. (3a)^3
b. (5.6)^2
c. (xy)^5
Answer :
a. (3a)^3 = 3^3 x a^3 = 27a^3
b. (5.6)^2 = 5^2 x 6^2 = 25 x 30 = 900
c. (xy)^5 = x^5 x y^5
In general number ;
If a is real number and m is positive integer, so :
(ab)^m = a^m x b^m
Characteristics 5 (Exponen of Divide Result of Two Numbers)
If division of two numbers is exponented so eachs number must be exponented.
Example 1.5 :
Determine exponent result bellow :
1. (1/6)^2
2. (3/4)^3
3. (a/7)^2
Answer :
1. (1/6)^2 = 1/6 x 1/6 = 1/36 = 1^2 / 6^2
2. (3/4)^3 = 3^3 / 4^3 = 27/64
3. (a/7)^2 = a^2 / 7^2 = a^2/49
In general form :
If a, b are real number and m is positive integer, so ;
(a/b)^m = a^m / b^m
with b ≠ 0
Exercise : 1.1
1. Simplify form bellow with using characteristics exponential number !
a. 3^2.3^3
b. 10^4.10^2
c. 8^3 : 8
d. 4^7 : 4^6
e. (2^4)^2
f. (5^3)^3
g. (2x)^4
h. (2^2.x^3.y)^3
2. Simplify form exponential bellow !
a. y^3.y^6
b. p^5 / p^3
c. (n^2)^8
d. (9^x)^3
e. b^6.b^2.b
f. e^2.e^m
g. 6^m.6^n
h. (pq)^r
i. (3a)^2
j. 7^x.7^y.7^c
k. (8^d)^2
l. e^5 : e^4
m. w^2 . w^10 : w^7
n. k^3 : k^2 : k
o. 3^c :3^d.3^2
p. 10^2.10^n
3. Simplify form bellow with using characteristics exponential number !
a. P^3.pq
b. S^3t^4.s^3t
c. m^2n / mn
d. 2k^10.3k^10
e. d^7.d^8 / d
f. 12v^6w : 4v^2
g. 4r^2t^5 / 2rt^2
h. 5xy.5^2
i. 3ab : 3^a
j. y^2z / 2.4y^3z / y
k. (3s^4r^3.2s^5t) / 8sr^3
l. h^n.h^n+1 / h^n-1
4. Write form bellow to simplest form !
a. (p^2q^2)
b. 64y^8 / (2y)^5
c. (b^3d^5)^2 / (bd^2)^3
d. 10(r^4s)^3 : (5r^2)^2
e. 10(r^4s)^3 : (5r^2)^2
f. (e(ef)^2)^3 / f
g. (3c)^4(ce)^2
h. (w^3x^2)(2wx)^2
i. (4m^2)(mn)^3 / 8m^2m
j. (v^5z)^2vz^5 / (vz)^3v^6z^3
k. (3x)^y (2x)^y
l. (a^m)^m+1 : (a^m-1 )^m
B. Exponential of Negative Integerand Nought
1. Exponential of Negative Integer
At characteristics 2, is known that a^m : a^n = a^m-n.
It only means that, if m>n. Now, we concern in this form.
a^3 / a^5 = a x a x a / a x a x a x a x a = 1/a^2; while a^3/a^5 = a^3-5
= a^-2 so, form 1/a^2 = a^-2 (form exponential negative integer)
In general form :
If a is negative integer, so a^-m = 1/a^m and 1/a^-m = a^m
Example 1.6 :
Change in to form positive exponential !
a. 2^-3
b. 1/10^-4
c. 2a^-n; a≠0
d. (a/b)^-m; a≠0, b≠0
Answer :
a. 2^-3 = 1/2^3 = 1/8
b. 1/10^-4 = 10^4 = 10000
c. 2a^-n = 2 . 1/a^n = 2/a^n
d. (a/b)^-m = 1/a^m / b^m = b^m / a^m = (b/a)^m
2. Nought Integer
If the characteristics 2, a^m : a^n = a^m-n, is exponended for m= so we got a^n : a^n = a^n-n = a^0
because a^n : a^n = a^n/a^n = a x a x……x a / a x a x….x a =1
so a^0 = 1
In general form :
If a is real number, and a≠0, so a^0 = 1
Example 1.7 :
Determine exponen result bellow !
a. 5^0
b. (-4)^0
c. (3/4)^0
d. (5q)^0; q≠0
Answer :
a. 5^0 = 1
b. (-4)^0 = 1
c. (3/4)^0 = 3^0 / 4^0 = 1/1 = 1
d. (5q)^0; q≠0 = 5^0 . q^0 = 1.1 = 1
Exercise 1.3 :
1. Simplify form exercise bellow become exponential of positive
a. 5^-1
b. 6^-2
c. 3^-2
d. 2^-4
e. 10^-3
f. (1/2)^-2
g. (1/7)^-1
h. (4/5)^-1
i. 2^-1/3^-1
j. 2^-1 . 4^-2
k. (0,5)^-1
l. (0,4)^-2
m. (0,5)^-2 / 5^-1
n. (2^2 . 4^-1)(2^-2 . 4^3)
o. 1 : 2^-2 : 3^-3
2. Determine result at form bellow !
a. 1^0
b. 3^0
c. (2/5)^0
d. (5m)^0
e. (xy)^0
f. 8q^0
g. (v^6)^0
h. (-5)^0
i. –(-x)^0
j. (k^0 + k^1)
k. 8^0 / 9^0
l. 3^0 . 0
3. Write form bellow to form exponential integer positive
a. q^-3
b. 2/c^-6
c. C^-3/d^-2
d. 2v/v^-1
e. t/pt^-4
"Rational Exponen, Form Of Root, and Logarithm"
Realization of base Competence which is showed with the study result bellow.
1. Understanding and using characteristic and rule of exponen, root, and logarithm in solving problem
2. Doing algebra manipulation in technical which is related to exponen, root, and logarithm
Rainbow is nature indication which is seen at the pouring rain. When sun in our back, length of wave and frequency of rainbow color are different. Try to find out and connected with material in this chapter!
You have heard about exponential number, right?Do you know, many cases in our life which is notatined in exponential number. For example length of wave and frequency of rainbow's color. Purple wave length of rainbow is 3,9 x 10^-7 metre until 7,7 x 10^14 metre and the frequency 6,9 x 10^14 Hertz until 7,7 x 10^14 Hertz. Another example, the length of DNA string (deoxcyribonuleic acid) in a cell 10^-7 metre and average humans body consist of 10^4 cells. DNA is part of main cell which involve in protein synthetic and genetic factor. The number of 10^-7 and 10^14 are exponen numbers. Try to find out the other examples.
A. Exponential of Positive Integer
In addition there is a process sum of n-th times can be written as
3 + 3 + 3 + 3 + 3 = 5 x 3
in other case of multiplication, there is a multiplication of n-th time can be written as
3 x 3 x 3 x 3 x 3 = 3^5
3^5 is called exponential number, 3 is called main of number and 5 is called exponen.
3^5 is called three to the power of five
in general form :
if a is real number and n is positive integer number, so :
a^n = a x a x a x …x a → n times factor
a^n read “ a exponen n “ with a is called main of number and n is called exponen.
Characteristics of exponential number with positive integer exponen.
Characteristics 1 (Multiplication of Exponential Number) :
If two exponential number or more that have same main of number, so the exponen must be sum.
Example 1.1 :
Determine multiplication result bellow :
a. 5^3 x 5^4
b. 5 x 5^3
c. 13^2 x 13^5 x 13
Answer :
a. 5^3 x 5^4 = 5 (3+4) = 5 7
b. 5 x 5^3 = 5(1+3) 5 4
c. 13^2 x 13^5 x 13 = 13(2+5+1) = 138
In general form :
If a is real number and m, n are positive integer, so :
a^m x a^n = a(m+n)
Characteristics 2 (Division of Exponential Number)
If an exponential number is divided to the other exponen number that have same main of number so the exponen have to be subtracted.
Example 1.2 :
Determine division result bellow :
1. 2^3 : 2
2. 5^6 : 5^4
3. 8^7 : 8^7
Answer :
1. 2^3 : 2 = 2 (3-1)
= 2^2
= 4
2. 5^6 : 5^4 = 5(6-4)
= 5^2
= 25
3. 8^7 : 8^2 = 8(7-2)
= 8^2
In general form :
If a is real number and m,n are positive integer, so :
a^m – a^n = a(m-n)
with a ≠ 0, m > 0
Characteristics 3 (Exponen of Exponential Number)
If an exponential number exponentized to the other number, so the exponen must be multiplicated.
Example 1.3 :
Determine exponen result bellow :
1. (3^3)^2
2. (5^2)^4
3. (10^3)^7
Answer :
1. (3^3)^2 = 3^3x2
= 3^6
2. (5^2)^4 = 5^2x4
= 5^8
3. (10^3)^7 = 10^3x7
= 10^21
In general form :
If a is real number and m,n are positive integer, so :
(a^m)^n = a^mxn
Characteristics 4 (Exponen of Multiplication Number)
If the multiplication of two number or more be exponented, so eachs number must be exponented,
Example 1.4 ;
Determine exponent result bellow ;
a. (3a)^3
b. (5.6)^2
c. (xy)^5
Answer :
a. (3a)^3 = 3^3 x a^3 = 27a^3
b. (5.6)^2 = 5^2 x 6^2 = 25 x 30 = 900
c. (xy)^5 = x^5 x y^5
In general number ;
If a is real number and m is positive integer, so :
(ab)^m = a^m x b^m
Characteristics 5 (Exponen of Divide Result of Two Numbers)
If division of two numbers is exponented so eachs number must be exponented.
Example 1.5 :
Determine exponent result bellow :
1. (1/6)^2
2. (3/4)^3
3. (a/7)^2
Answer :
1. (1/6)^2 = 1/6 x 1/6 = 1/36 = 1^2 / 6^2
2. (3/4)^3 = 3^3 / 4^3 = 27/64
3. (a/7)^2 = a^2 / 7^2 = a^2/49
In general form :
If a, b are real number and m is positive integer, so ;
(a/b)^m = a^m / b^m
with b ≠ 0
Exercise : 1.1
1. Simplify form bellow with using characteristics exponential number !
a. 3^2.3^3
b. 10^4.10^2
c. 8^3 : 8
d. 4^7 : 4^6
e. (2^4)^2
f. (5^3)^3
g. (2x)^4
h. (2^2.x^3.y)^3
2. Simplify form exponential bellow !
a. y^3.y^6
b. p^5 / p^3
c. (n^2)^8
d. (9^x)^3
e. b^6.b^2.b
f. e^2.e^m
g. 6^m.6^n
h. (pq)^r
i. (3a)^2
j. 7^x.7^y.7^c
k. (8^d)^2
l. e^5 : e^4
m. w^2 . w^10 : w^7
n. k^3 : k^2 : k
o. 3^c :3^d.3^2
p. 10^2.10^n
3. Simplify form bellow with using characteristics exponential number !
a. P^3.pq
b. S^3t^4.s^3t
c. m^2n / mn
d. 2k^10.3k^10
e. d^7.d^8 / d
f. 12v^6w : 4v^2
g. 4r^2t^5 / 2rt^2
h. 5xy.5^2
i. 3ab : 3^a
j. y^2z / 2.4y^3z / y
k. (3s^4r^3.2s^5t) / 8sr^3
l. h^n.h^n+1 / h^n-1
4. Write form bellow to simplest form !
a. (p^2q^2)
b. 64y^8 / (2y)^5
c. (b^3d^5)^2 / (bd^2)^3
d. 10(r^4s)^3 : (5r^2)^2
e. 10(r^4s)^3 : (5r^2)^2
f. (e(ef)^2)^3 / f
g. (3c)^4(ce)^2
h. (w^3x^2)(2wx)^2
i. (4m^2)(mn)^3 / 8m^2m
j. (v^5z)^2vz^5 / (vz)^3v^6z^3
k. (3x)^y (2x)^y
l. (a^m)^m+1 : (a^m-1 )^m
B. Exponential of Negative Integerand Nought
1. Exponential of Negative Integer
At characteristics 2, is known that a^m : a^n = a^m-n.
It only means that, if m>n. Now, we concern in this form.
a^3 / a^5 = a x a x a / a x a x a x a x a = 1/a^2; while a^3/a^5 = a^3-5
= a^-2 so, form 1/a^2 = a^-2 (form exponential negative integer)
In general form :
If a is negative integer, so a^-m = 1/a^m and 1/a^-m = a^m
Example 1.6 :
Change in to form positive exponential !
a. 2^-3
b. 1/10^-4
c. 2a^-n; a≠0
d. (a/b)^-m; a≠0, b≠0
Answer :
a. 2^-3 = 1/2^3 = 1/8
b. 1/10^-4 = 10^4 = 10000
c. 2a^-n = 2 . 1/a^n = 2/a^n
d. (a/b)^-m = 1/a^m / b^m = b^m / a^m = (b/a)^m
2. Nought Integer
If the characteristics 2, a^m : a^n = a^m-n, is exponended for m= so we got a^n : a^n = a^n-n = a^0
because a^n : a^n = a^n/a^n = a x a x……x a / a x a x….x a =1
so a^0 = 1
In general form :
If a is real number, and a≠0, so a^0 = 1
Example 1.7 :
Determine exponen result bellow !
a. 5^0
b. (-4)^0
c. (3/4)^0
d. (5q)^0; q≠0
Answer :
a. 5^0 = 1
b. (-4)^0 = 1
c. (3/4)^0 = 3^0 / 4^0 = 1/1 = 1
d. (5q)^0; q≠0 = 5^0 . q^0 = 1.1 = 1
Exercise 1.3 :
1. Simplify form exercise bellow become exponential of positive
a. 5^-1
b. 6^-2
c. 3^-2
d. 2^-4
e. 10^-3
f. (1/2)^-2
g. (1/7)^-1
h. (4/5)^-1
i. 2^-1/3^-1
j. 2^-1 . 4^-2
k. (0,5)^-1
l. (0,4)^-2
m. (0,5)^-2 / 5^-1
n. (2^2 . 4^-1)(2^-2 . 4^3)
o. 1 : 2^-2 : 3^-3
2. Determine result at form bellow !
a. 1^0
b. 3^0
c. (2/5)^0
d. (5m)^0
e. (xy)^0
f. 8q^0
g. (v^6)^0
h. (-5)^0
i. –(-x)^0
j. (k^0 + k^1)
k. 8^0 / 9^0
l. 3^0 . 0
3. Write form bellow to form exponential integer positive
a. q^-3
b. 2/c^-6
c. C^-3/d^-2
d. 2v/v^-1
e. t/pt^-4
Jumat, 02 Januari 2009
“The definition, explanation, and the examples of the terms questioning by Ardhita Kirana Rukmi”
“The definition, explanation, and the examples of the terms questioning
by Ardhita Kirana Rukmi”
1. Besaran
In English : Value
Explanation : Expression of arithmetic, algebra, or analytic that more related with value than relation between that expression]
Example : Unit of long value is metre
2. Jarak Tempuh
In English : Travelled Distance
Explanation : The distance that used to go through journey
Example : Travelled distance from Yogyakarta to Kulon Progo is 21 km
3. Waktu yang diperlukan untuk menempuh jarak
In English : Time that is need to follow the distance
Example : Time that is need to follow the distance as far as 300 matres is 15 minuts
4. Lingkaran dalam segitiga dan lingkaran luar segitiga
In English : Circle-in the triangle and the outer circle
Explanation of Circle-in the triangle : The circle that touch on the three sides of triangle
Explanation of the outer circle : The circle that trough three angle points of a triangle
Example : Circle with center P is circle in the triangle
5. Sudut dalam dan sudut luar
In English : Interior angle of triangle and exterior angle of triangle
Explanation of Interior angle of triangle : the angle that the measurement of his angle is smaller than the measurement of the angle of this triangle
Explanation of exterior angle of triangle : The angle that his measurement is bigger than the measurement of the angle of this triangle
6. Perbedaan antara luas bidaang dan luas daerah
In English : The different between plane anf area
Example : The are a of trapezium is 96 cm^2
The plane of rectangle is 100 cm^2
7. Himpunan Bagian
In English : Subset
Explanation : A part from group of things of the same kind that belong together
8. Membedakan segitiga sama kaki, sama sisi, dan segitiga sembarang
In English : Difference the isosceles, equilateral, and scalene triangle
Explanation of isosceles triangle : Triangle having two sides equal in length
Explanation of equilateral triangle: Triangle having three sides equal in length
Explanation of scalene triangle : Triangle having all sides not equal in length
Example : Triangle ABC is isosceles triangle
Triangle PQR is equilateral triangle
Triangle KLM is scalene triangle
9. Sudut Bagi
In English : Devide angle
Explanation : Angle between two plane
Example : Angle B is a devide angle of triangle ABC
10. Berapa waktu yang diperlukan roda sebuah sepeda motor untuk berputar selama 300 kali
In English : How much time that is need the wheel of a motorcycle to proce fore 300 times
11. Menentukan sudut keliling dalam-yang menghadap busur
In English : Determine inscribed angle that intercept art
by Ardhita Kirana Rukmi”
1. Besaran
In English : Value
Explanation : Expression of arithmetic, algebra, or analytic that more related with value than relation between that expression]
Example : Unit of long value is metre
2. Jarak Tempuh
In English : Travelled Distance
Explanation : The distance that used to go through journey
Example : Travelled distance from Yogyakarta to Kulon Progo is 21 km
3. Waktu yang diperlukan untuk menempuh jarak
In English : Time that is need to follow the distance
Example : Time that is need to follow the distance as far as 300 matres is 15 minuts
4. Lingkaran dalam segitiga dan lingkaran luar segitiga
In English : Circle-in the triangle and the outer circle
Explanation of Circle-in the triangle : The circle that touch on the three sides of triangle
Explanation of the outer circle : The circle that trough three angle points of a triangle
Example : Circle with center P is circle in the triangle
5. Sudut dalam dan sudut luar
In English : Interior angle of triangle and exterior angle of triangle
Explanation of Interior angle of triangle : the angle that the measurement of his angle is smaller than the measurement of the angle of this triangle
Explanation of exterior angle of triangle : The angle that his measurement is bigger than the measurement of the angle of this triangle
6. Perbedaan antara luas bidaang dan luas daerah
In English : The different between plane anf area
Example : The are a of trapezium is 96 cm^2
The plane of rectangle is 100 cm^2
7. Himpunan Bagian
In English : Subset
Explanation : A part from group of things of the same kind that belong together
8. Membedakan segitiga sama kaki, sama sisi, dan segitiga sembarang
In English : Difference the isosceles, equilateral, and scalene triangle
Explanation of isosceles triangle : Triangle having two sides equal in length
Explanation of equilateral triangle: Triangle having three sides equal in length
Explanation of scalene triangle : Triangle having all sides not equal in length
Example : Triangle ABC is isosceles triangle
Triangle PQR is equilateral triangle
Triangle KLM is scalene triangle
9. Sudut Bagi
In English : Devide angle
Explanation : Angle between two plane
Example : Angle B is a devide angle of triangle ABC
10. Berapa waktu yang diperlukan roda sebuah sepeda motor untuk berputar selama 300 kali
In English : How much time that is need the wheel of a motorcycle to proce fore 300 times
11. Menentukan sudut keliling dalam-yang menghadap busur
In English : Determine inscribed angle that intercept art
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