Transcript Slide 1

S.K.1. Memecahkan masalah yang
berkaitan dengan Konsep
Operasi Bilangan Real
K.D.2. Menerapkan Operasi pada
Bilangan Berpangkat
( Exponent )
Tujuan Pembelajaran :
1. Siswa dapat mengoperasikan
bilangan berpangkat
2. Siswa dapat menyederhanakan
bilangan berpangkat
EXPONENT
Rene Descartes(1550-1617)
A France mathematician,
introduced the method of
writing exponent for the first
time
MAIN TOPIC
Definition of The Positive Integer Exponent
In junior high school, you had learned about
exponential number with base 10. the positive
integer exponent implies how many copies of
the base are multiplied together
(Bilangan Berpangkat adalah suatu cara
perkalian dengan bilangan yang sama)
Example:
an = a x a x a x … x a , sebanyak n faktor
34 = 3 x 3 x 3 x 3
57 = 5 x 5 x 5 x 5 x 5 x 5 x 5
Definition of Exponent
a
b
if
a
=c
c b
b = cardinal number ( bilangan pokok )
a = exponent number ( bilangan pangkat )
c = exponent of the number ( bilangan hasil perpangkatan )
Example :
if
3
2
2
= 9
9 3 ,
then
2
9
=
9
The Properties of Exponent ( Formula )
1
a
xa a
m
n
2
a : a a
3
a 
m
4
n
m n
, a  0 and m  n
m xn
(axb)  a x b
n
m
5
a
m n
m n
n
a
a
   m,
b
b
m
n
with b  0
The Properties of Exponent
6
(a . b )  a .b
m
-
m
n
7
a
8
a

 bm

1

a
m
9
m n
a
m
n
m
n
n

a m. n

  b m. n

1

a

m
n
mn
mn
Example aplication formula
1. aⁿ = a . a . a ….. a , sebanyak n faktor
Contoh :
a³ = a . a . a
2³ = 2 . 2 . 2
2. aᵐ . aⁿ = aᵐ ⁺ ⁿ
Contoh :
a³ . a⁴ = a³ ⁺ ⁴ = a⁷
a⁶ . aˡ . a⁵ = a ⁶ ⁺ ˡ ⁺ ⁵ = aᴵ² , (a sebagai bilangan
pokok harus sama)
3.
aᵐ : aⁿ = aᵐ ̄ ⁿ
Contoh :
a⁸ : a² = a ⁸ ̄ ² = a⁶
a³ . b⁴ = a³ . a ̄ ⁵ . b ⁴ . b ̄ ⁷ = a³ ̄ ⁵ . b ⁴ ̄ ⁷
a⁵ . b⁷
= ā² . b̄³
4.
( aᵐ )ⁿ = aᵐ·n
Contoh :
( a³ )² = a³·² = a⁶
{(a³)²}⁴ = a³·²·⁴ = a²⁴
( a³ . b )⁴ = a³·⁴ . b ⁴ = aˡ² . b⁴
a ² ⁵ = aˡ⁰
b³
bˡ⁵
5. ᵐ√aⁿ = a n/m
Contoh :
⁵√a³ = a3/5
√a = ²√aˡ = a1/2
√x = x1/2
x² . √x = x² . X 1/2 = x²⁺1/2 = x5/2
6. a ̄ ⁿ = 1/aⁿ
3 ̄ ² = 1/3² = 1/9
2 ̄ ³ = 1/2³ = 1/8
7. a⁰ = 1
10000 ⁰ = 1
, aⁿ
, 3²
, 2³
=
x² 1/2
= 1/a ̄ ⁿ
= 1/3 ̄ ²
= 1/2 ̄ ³
Example 1:
Simplify the following expressions!
a. ((6a2b3)2)4
b. (23a2b3)4 x (2ab2)3
Answer :
a. ((6a2b3)2)4 = (62.a2x2.b3x2)4
= (62.a4.b6)4
= 68.a16.b24
b.
(23a2b3)4 x (2ab2)3
Answer :
= (23a2b3)4 x (2ab2)3
= (23x4 . a2x4 . b3x4) x (23 . a3 . b2x3)
= (212 . a8. b12) x (23 . a3 . b6)
= (212+3 . a8+3 . b12+6)
= 215. a11 . b18
Example 2:
Simplify and state each of the
following expressions in their positive
integer exponents!
a. 2p3q-4
b. a-7b5c-9 : 10-10c7d-6
c. (5-2m2n-5)-4
Answer :
1.
2.
2p3q-4
3
= 2P
q4
a-7b5c-9 : 10-10c7d-6
a 7b 5 c 9
=
1 010 c 7 d 6
1 010 b 5 d6
=
a 7 c 16
3.
(5-2m2n-5)-4
= 5-2.-4 . m2.-4 . n-5.-4
= 58 m-8 n20
5 8 n 20
=
m8
Competence Check:
1. Simplify the following expressions!
a. ((-6a2b3)2)4
b.
(23a2b3)4 x (2ab2)3
c.
 2 p q  p q3
 2  x 
2
 p q   pq



2
2. Simplify and state each of the following expressions
in their positive integer exponents!
a.  7x2  1  7x1  3

 . 

 y .x   y 
1
2 2
-3



3p
q
r
b.  2  :  6 
 r 
p 


 
c. (5a2b-3)-3 . 3(a2b3)2
ROOT
There are so many phenomena in our
life which Can be modeled to the
function or equation containing roots.
Let start our discussion about concept
of roots by studying the rational and
irrational number first
Definition of Rational and Irrational
Numbers
Rational numbers are
numbers that can be
expressed as fraction
a/b, where a and b
are integers and b 0
Definition of Root
Root are numbers in the root symbol
which cannot produce rational
numbers
Example :
,
10
17:05:48
7
Algebra Operation Of The Roots
a. Addition and Subtraction of the roots
a c  b c (a
 b) c
b. Multiplication of Roots
There are several properties of multiplication of
roots, such as:
1.
2.
c x d  cxd
3.
a c x b d  ab c x d
a( c  b d )  a c  ab d
Example:
1. Study the following addition and subtraction
a. 4 3  2 3
b. 4 5  2 5
2. Simplify there following expression
a. 2 x 12
b.

5 3

5 3

Answer :
1. a.
b.
2. a.
b.
4 3  2 3  4  2 3  6 3
4 5  2 5  (4  2) 5  2 5

2 x 12  2x12  24  2 6
5 3

5 3
=
5x 5 
= 5  15 
= 8  2 15

5x 3 
15  3
3x 5 
3x 3
Simplifying the Form of :
 p  q  2
pq 
p q
 p  q  2
pq 
p q
Example :
Simplify each of the following roots!
a.
8  2 15
b.
6  32
Answer :
a.
b.
8  2 15  (5  3)  2 5.3
 5 3
6  32  6  2 8
 (4)  2 4.2
 4 2
Rationalizing the Denominator of Fraction:
a
a
a.

x
b
b
b a b

b
b
a
a
b  c ab a c
b.

x

bc
b c b c
b c
a
a
c.

x
b c
b c
b c a b c

bc
b c
b c
b c
d.

x
b c
b c
b c

b c


b c b c
bc

Thank You