The document discusses exponents and roots in mathematics. It provides definitions and properties of exponents, including laws of exponents for positive, negative, and fractional exponents. Examples are given of exponent operations and simplifying expressions with exponents. Roots are then introduced, defining them as the inverse of exponents. Properties of roots are covered, along with examples of simplifying and rationalizing root expressions through operations.

1. BENTUK EKSPONEN
& BENTUK AKAR
7th grade

2. BENTUK EKSPONEN
(PANGKAT)
01

3. Bilangan Berpangkat Bilangan Bulat
Bilangan bulat terdiri dari bilangan bulat yang memiliki nilai
positif, bilangan bulat dengan nilai negatif, dan nol. Maka
dapat disumpulkan bahwa bilangan berpangkat bilangan
bulat adalah bilangan-bilangan yang berpangkat positif,
negatif, dan nol.

4. Sifat – sifat eksponen
1. Bilangan Berpangkat 0
Untuk bilangan bulat dengan pangkat 0, hasilnya
adalah 1. Jadi, bilangan bulat apapun itu baik itu
nilainya negatif atau positif, jika dipangkatkan dengan
0 maka hasilnya adalah 1, tapi ini tidak berlaku untuk
bilangan bulat 0.

5. Sifat – sifat eksponen
2. Bilangan Berpangkat Bulat Positif
Beberapa sifat dari bilangan berpangkat bulat positif,
diantaranya adalah sebagai berikut ini:
amx an = am+n
am : an = am-n , untuk m>n dan b ≠ 0
(am)n = amn
(ab)m = am bm
𝑎
𝑏
𝑚
=
𝑎𝑚
𝑏𝑚 , untuk b ≠ 0

6. Sifat – sifat eksponen
3. Bilangan Berpangkat Bulat Negatif
Untuk sifat bilangan berpangkat bulat negatif
adalah:
Jika a∈R, a ≠ 0, dan n merupakan bilangan
bulat negatif, maka:
a-n =
1
𝑎𝑛 atau an =
1
𝑎−𝑛

7. Sifat – sifat eksponen
4. Bilangan Berpangkat Pecahan
• 𝑎
1
𝑛= 𝑛
𝑎
• 𝑎
𝑚
𝑛 =
𝑛
𝑎𝑚= 𝑛
𝑎 𝑚

8. Sifat – sifat eksponen
Namun, ada sebuah pengecualian untuk kasus bilangan pokok yang
berpangkat negatif. Ada beberapa poin yang harus kamu ketahui :
 Bilangan negatif pangkat genap = Hasilnya positif
 Bilangan negatif pangkat ganjil = Hasilnya negatif

9. Operasi Hitung yang Melibatkan
Bilangan Berpangkat
Ada beberapa hal yang harus diperhatikan sebelum kamu mengerjakan
soal operasi hitung yang melibatkan bilangan berpangkat, antara lain:
1. Kerjakan operasi dalam kurung terlebih dahulu
2. Lanjutkan dengan operasi perpangkatan
3. Kerjakan operasi perkalian dan pembagian
4. Kerjakan operasi penjumlahan atau pengurangan.

10. Perkalian Pada Perpangkatan
Pada operasi hitung perkalian dalam bilangan berpangkat, berlaku sifat
seperti di bawah ini:
Untuk lebih memahami tentang perkalian pada perpangkatan, perhatikan
contoh berikut:
63 x 62 = (6 x 6 x 6) x (6 x 6)
63 x 62 = 6 x 6 x 6 x 6 x 6
63 x 62 = 65
Sehingga dapat kita simpulkan menjadi 63 x 62 = 62+3 = 65
am x an = am+n

11. Contoh soal :
Tentukan nilai dari 23 x 24 dan 52 x 52 :
Penyelesaian :
 23 x 24 = 2 3+4 = 27 = 128
 52 x 52 = 5 2+2 = 5 4 = 625
Tentukan nilai dari
54
53 dan
78
76 :
Penyelesaian :

54
53 = 54-3 = 51 = 5

78
76 = 78-6 = 72= 49

12. Contoh soal :
Tentukan nilai dari (2x3)2 dan 53x 23
Penyelesaian :
 (2 x 3)2 = 22x32 = 4 x 9 = 36
 53x 23 = (5 x 2)3 = 103 = 1000
Tentukan nilai dari (32)2:
Penyelesaian :
(32)2 = 32x2 =34 =81

13. Contoh soal :
Tentukan nilai dari 50 dan −
1
3
0
Penyelesaian :
 50 = 1
 −
1
3
0
= 1
Tentukan nilai dari
2
5
2
dan
63
23 :
Penyelesaian :

2
5
2
=
22
52 =
4
25

63
23 =
6
2
3
= 33 = 27

14. Contoh soal :
Tentukan nilai dari 3
2
3 dan 16
3
4 :
Penyelesaian :
 3
2
3 =
3
32 =
3
9
 16
3
4 =
4
16
3
= 23= 8 ; atau
 16
3
4 = 24
3
4 = 24 𝑥
3
4 = 23= 8
Tentukan nilai dari (-2)4 dan −2 5:
Penyelesaian :
 (-2)4 = 24 = 16
 −2 5 = -(25) = -32

15. Contoh soal :
Tentukan nilai dari 2-3 ,
1
2−2 dan
3
2−4:
Penyelesaian :
 2-3 =
1
23 =
1
8

1
2−2 = 22 = 4

3
2−4 = 3 x 24 = 3 x 16 = 48
Tentukan nilai dari 4
1
2 dan 3
1
5 :
Penyelesaian :
 4
1
2 =
2
4 = 2
 3
1
5 =
5
3

16. Bentuk Akar
02

17. Bentuk Akar
Pada materi sebelumnya telah dibahas tentang sifat-sifat
eksponen yang diantaranya dengan pangkat pecahan. Bentuk
akar ada hubungannya dengan pangkat pecahan. Berikut bentuk
pangkat pecahan yang digunakan :
Jika 𝑛
𝑎 = 𝑏 maka, 𝑎 = 𝑏𝑛

18. Bentuk Akar
Catatan :
 𝑛
𝑎 hasilnya 𝑏 yang memenuhi 𝑏𝑛
= 𝑎
 𝑛
𝑎 dibaca akar pangkat 𝑛 dari 𝑎
 Khusus untuk 𝑛 = 2, 2
𝑎 cukup ditulis 𝑎 dan dibaca akar dari 𝑎
atau akar kuadrat dari 𝑎 atau akar 𝑎
𝑝 = 𝑞 ↔ 𝑞2
= 𝑝
Bentuk Akar

19. Pengertian Bentuk Akar
Bentuk akar adalah akar dari sebuah bilangan real positif yang hasilnya
bukan bilangan rasional yang memenuhi sifat :
Jika 𝑎 = 𝑏 maka 𝑏2
= 𝑎 dengan 𝑎 ≥ 0
Catatan :
 𝑏 adalah hasil dari 𝑎
 𝑎 disebut bentuk akar jika hasilnya (𝑏) adalah bilangan irrasional
Contoh :
1. 4 bukan bentuk akar, karena hasil 4 = 2 adalah bilangan rasional
2. 2 adalah bentuk akar karena hasilnya 2 = 1,41421…. adalah bilangan irrasional

20. Menyederhanakan Bentuk Akar
Untuk menyederhanakan bentuk akar, kita gunakan sifat 𝑎2 = 𝑎 ,
dengan 𝑎2
disebut sebagai bilangan kuadrat sempurna, serta gunakan
sifat 𝑎2. 𝑏 = 𝑎 𝑏 dengan 𝑎 ≥ 0 dan 𝑏 ≥ 0
Contoh
1) 4 = 22 = 2
2) 32 = 16 . 2 = 42 . 2 = 4 2
3) 𝑐3 = 𝑐2. 𝑐 = 𝑐 𝑐
4) 12𝑘 = 4 . 3𝑘 = 22 . 3 𝑘 = 2 𝑘
5) 18𝑘3 = 9𝑘2. 2𝑘 = 3𝑘 2 . 2𝑘 = 3𝑘 2𝑘

21. Operasi Aljabar Bentuk Akar
1) 𝑎 𝑝 + 𝑏 𝑝 = 𝑎 + 𝑏 𝑝
2) 𝑎 𝑝 − 𝑏 𝑝 = 𝑎 − 𝑏 𝑝
3) 𝑎 . 𝑏 = 𝑎 . 𝑏
4)
𝑎
𝑏
=
𝑎
𝑏
5) 𝑎 𝑝 . 𝑏 𝑞 = 𝑎 . 𝑏 𝑝 . 𝑞
6) 𝑎 . 𝑎 = 𝑎2 = 𝑎
7)
𝑎 𝑝
𝑏 𝑞
=
𝑎
𝑏
𝑝
𝑞

22. Operasi Aljabar Bentuk Akar
Contoh :
1. 2 3 + 4 3 = 2 + 4 3 = 6 3
2. 6 5 − 2 5 = 6 − 2 5 = 4 5
3. 2 . 3 = 2 . 3 = 6
4.
6
3
=
6
3
= 2
5. 3 5 . 2 2 = 3 . 2 5 . 2 = 6 10
6. 7 . 7 = 72 = 49 = 7
7.
8 15
2 3
=
8
2
15
3
= 4 5

23. Merasionalkan Bentuk Akar
Merasionalkan bentuk akar adalah mengubah bentuk akar (irasional) menjadi bilangan
rasional (menghilangkan akarnya) dengan mengalikan bentuk sekawannya.
Untuk a,b,c dan d bilangan rasional positif, maka :
 𝑎 sekawan dengan 𝑎
 𝑎 + 𝑏 sekawan dengan 𝑎 − 𝑏 (begitupun sebaliknya)
 𝑎 + 𝑝 𝑏 sekawan dengan 𝑎 − 𝑝 𝑏 (begitupun sebaliknya)
 𝑎 + 𝑏 sekawan dengan 𝑎 + 𝑏 (begitupun sebaliknya)
 𝑝 𝑎 + 𝑞 𝑏 sekawan dengan 𝑝 𝑎 − 𝑞 𝑏 (begitupun sebaliknya)

24. Contoh :
Rasionalkan bentuk pecahan-pecahan dibawah ini :
a)
2
3
b)
5
4− 6
c)
14
5 + 3
d)
2
5−2 3
e)
3
2 5 − 3 2
Jawab :
a)
2
3
=
2
3
.
3
3
=
2 3
3
=
2
3
3

25. Jawab :
b)
5
4− 6
=
5
4− 6
.
4+ 6
4+ 6
=
5 (4+ 6)
16−6
=
5 (4+ 6)
10
=
4+ 6
2
c)
14
5+ 3
=
14
5+ 3
.
5− 3
5− 3
=
14 ( 5− 3)
5−3
=
14 ( 5− 3)
2
= 7 5 − 3
d)
2
5−2 3
=
2
5−2 3
.
5+2 3
5+2 3
=
2(5+2 3 )
25−(4.3)
=
2(5+2 3 )
13
e)
3
2 5 − 3 2
=
3
2 5 − 3 2
.
2 5 + 3 2
2 5 + 3 2
=
3 (2 5 + 3 2)
4.5 −9.2
=
3 (2 5 + 3 2)
2

26. Bentuk Akar dalam Akar
Untuk 𝑎 dan 𝑏 bilangan rasional positif, maka berlaku sifat :
 𝑎 + 𝑏
2
= 𝑎 + 𝑏 + 2 𝑎𝑏 atau
𝑎 + 𝑏 + 2 𝑎𝑏 = 𝑎 + 𝑏
 𝑎 − 𝑏
2
= 𝑎 + 𝑏 − 2 𝑎𝑏 atau
𝑎 + 𝑏 − 2 𝑎𝑏 = 𝑎 − 𝑏 dengan 𝑎 ≥ 𝑏

27. Contoh :
a) 4 + 2 3
b) 6 − 2 8
c) 7 + 24
d) 4 + 15
Jawab :

28. 1
Measurement units
Inch
Neptune is very far away
from Earth
Foot
Saturn is a gas giant
and has several rings
Mercury is the closest
planet to the Sun
Mile
Yard
Despite being red, Mars
is a cold place
63360
5280
1760

29. Units of measurement in our daily life
Mercury is the closest
planet to the Sun
Mercury
It has a beautiful name,
but it’s terribly hot
Venus
Despite being red,
Mars is a cold place
Mars
It’s the biggest planet
in the Solar System
Saturn is the ringed
one and a gas giant
It’s the farthest planet
from the Sun
Jupiter Saturn Neptune

30. 01
Table of contents
Concepts
You can describe the topic
of the section here
Exercises
03
You can describe the topic
of the section here
Conclusions
04
You can describe the topic
of the section here
02
You can describe the topic
of the section here
Introduction

31. You can't forget this
Despite being red,
Mars is actually a very
cold place
Mars
It’s a gas giant and
the biggest planet in
the Solar System
It’s the ringed planet,
composed mostly of
hydrogen and helium
Saturn
Jupiter

32. 01
Table of contents
Concepts
You can describe the topic
of the section here
Exercises
03
You can describe the topic
of the section here
Conclusions
04
You can describe the topic
of the section here
02
You can describe the topic
of the section here
Introduction

33. —Someone Famous
“This is a quote, words full of
wisdom that someone
important said and can make
the reader get inspired.”

34. Whoa!!
This can be the part of the presentation where you
introduce yourself, write your email...

35. Let's talk about units of measurement
Do you know what helps you make your point clear?
Lists like this one:
● They’re simple
● You can organize your ideas clearly
● You’ll never forget to buy milk!
And the most important thing: the audience won’t miss
the point of your presentation

36. 10,000,000
Big numbers catch your audience’s attention

37. Awesome words
Because key words are great for catching your audience’s attention

38. A picture is worth
a thousand words

39. Change of units
Mercury is the
smallest planet
Inch
Despite being red,
Mars is a cold place
Yard
Saturn is a gas
giant and has rings
Foot
95004 7917 2639 1,5
Jupiter is the biggest
planet of them all
Mile

40. Multiplication table
Table 1 Table 2 Table 3 Table 4 Table 5
1x1= 1 1x6= 6 2x1= 2 2x6= 12 3x1= 3 3x6= 18 4x1= 4 4x6= 24 5x1= 5 5x6= 30
1x2= 2 1x7= 7 2x2= 4 2x7= 14 3x2= 6 3x7= 21 4x2= 8 4x7= 28 5x2= 10 5x7= 35
1x3= 3 1x8= 8 2x3= 6 2x8= 16 3x3= 9 3x8= 24 4x3= 12 4x8= 32 5x3= 15 5x8= 40
1x4= 4 1x9= 9 2x4= 8 2x9= 18 3x4= 12 3x9= 27 4x4= 16 4x9= 36 5x4= 20 5x9= 45
1x5= 5 1x10= 10 2x5= 10 2x10= 20 3x5= 15 3x10= 30 4x5= 20 4x10= 40 5x5= 25 5x10= 50

41. Multiplication table
Table 6 Table 7 Table 8 Table 9 Table 10
6x1= 6 6x6= 36 7x1= 7 7x6= 42 8x1= 8 8x6= 48 9x1= 9 9x6= 54 10x1= 10 10x6= 60
6x2= 12 6x7= 42 7x2= 14 7x7= 49 8x2= 16 8x7= 56 9x2= 18 9x7= 63 10x2= 20 10x7= 70
6x3= 18 6x8= 48 7x3= 21 7x8= 56 8x3= 24 8x8= 64 9x3= 27 9x8= 72 10x3= 30 10x8= 80
6x4= 24 6x9= 54 7x4= 28 7x9= 63 8x4= 32 8x9= 72 9x4= 36 9x9= 81 10x4= 40 10x9= 90
6x5=30 6x10= 60 7x5= 35 7x10= 70 8x5= 40 8x10= 80 9x5= 45 9x10= 90 10x5= 50
10x10=
100

42. How to calculate the volume?
Venus is the second
planet from the Sun
Width
Jupiter is the biggest
planet of them all
Height
Despite being red, Mars
is a cold place
Depth
1
2
3

43. 333,000.00
The Sun’s mass compared to Earth’s
9h 55m 23s
Jupiter’s rotation period
386,000 km
Distance between Earth and the Moon

44. Where are these units used?
Mercury is the
closest planet to
the Sun
It’s the biggest
planet in the Solar
System
It’s composed of
hydrogen and
helium
Saturn
Mercury
Jupiter
1
2
3
3 2
1

45. Exercises
You can enter a subtitle
here if you need it
03

46. Weight and measurement: exercise
Measure and weigh objects that have these shapes
Weight
?
Weight
30 lb
Weight
?
Length
4 in
Length
?
Length
?

47. Let's do some calculations
Solve these additions and subtractions
1. 10 + 4 =
2. 5 + 8 =
3. 12 + 5 =
4. 7 + 6 =
5. 8 + 7 =
6. 9 + 2 =
7. 8 + 5 =
8. 13 + 8 =
9. 12 + 7 =
10. 15 + 6=
Additions
1. 9 - 5 =
2. 15 - 4 =
3. 8 - 3 =
4. 16 - 7 =
5. 7 - 4 =
6. 19 - 8 =
7. 17 - 9 =
8. 16 - 4 =
9. 12 - 7 =
10. 11 - 8 =
Subtractions

48. Let's do some calculations
Solve these additions and subtractions
1. 10 + 4 = 14
2. 5 + 8 = 13
3. 12 + 5 = 17
4. 7 + 6 = 13
5. 8 + 7 = 15
6. 9 + 2 = 11
7. 8 + 6 = 14
8. 13 + 8 = 21
9. 12 + 7 = 19
10. 15 + 6= 21
Additions
1. 9 - 5 = 4
2. 15 - 4 = 11
3. 8 - 3 = 5
4. 16 - 7 = 9
5. 7 - 4 = 3
6. 19 - 8 = 11
7. 17 - 9 = 8
8. 16 - 4 = 12
9. 12 - 7 = 5
10. 11 - 8 = 3
Subtractions

49. Let's calculate the time
Determine the end time of each action
_ _:_ _
At ten o'clock at night
_ _:_ _
At eleven thirty in the morning
_ _:_ _
At twenty past seven in the afternoon
It started Takes Finished
_ _:_ _
_ _:_ _
_ _:_ _
35 min
15 min
45 min

50. Let's calculate the time
Determine the end time of each action
22:00
At ten o'clock at night
11:30
Half past eleven in the morning
19:20
Twenty past seven in the afternoon
It started Takes Finished
22:35
Twenty-five to eleven at night
11:45
Quarter to twelve in the morning
20:05
Five past eight in the afternoon
35 min
15 min
45 min

51. A picture always
reinforces the
concept
Use an image instead of a long text

52. Conclusions
You can enter a subtitle here
if you need it
04

53. Graph
Follow the link in the graph to modify its data and then paste the new one here. For more info, click here
Mercury is the closest
planet to the Sun
Mercury
Mars
Despite being red,
Mars is a cold place
Saturn
Saturn is composed of
hydrogen and helium
10k
20k
30k

54. Desktop
software
You can replace the image on
the screen with your own work

55. Tablet app
You can replace the image on
the screen with your own work

56. You can replace the image on
the screen with your own work
Mobile web

57. What about these percentages?
It’s the third planet
from the Sun
Earth
Despite being red,
Mars is a cold place
Mars
It’s the farthest planet
from the Sun
Neptune
50% 60%
25%

58. Our team
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