"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