Kurikulum Merdeka Matematika Kelas 8 Bab 1 Bilangan Berpangkat

Summary

This educational video provides a comprehensive overview of exponents in 8th-grade mathematics. It explains the concept of exponents, including how to multiply and divide numbers with exponents, the rules for raising a power to another power, and handling zero and negative exponents. The video also covers the conversion of exponents to and from radical form, simplifying radical expressions, and rationalizing denominators. It concludes with the importance of scientific notation and demonstrates how to convert large numbers into this standardized format, making complex numbers more manageable.

Takeaways

• 🔢 Exponents are a mathematical concept where a number is raised to a power, indicating the number of times the base is multiplied by itself.
• 📚 The script discusses the Indonesian curriculum for 8th-grade mathematics, focusing on the topic of exponents.
• 💡 When multiplying numbers with the same base, you can add the exponents to simplify the calculation, e.g., 3^2 * 3^3 equals 3^5.
• 🚫 If the bases are different, you must expand and multiply each term individually, such as 2^2 * 3^3 which equals 4 * 27.
• ➗ In division involving exponents, you subtract the exponent of the divisor from the exponent of the dividend, like 3^3 divided by 3^2 equals 3^1.
• 🔄 Raising a number to another power multiplies the exponents, for example, (3^3) squared equals 3^6.
• 📈 When raising a product to a power, you can raise each factor in the product to the power separately, e.g., (3 * 4) squared equals 3^2 * 4^2.
• 🎩 Any number raised to the power of zero equals one, regardless of the base, such as 1^0 = 1, 20^0 = 1, and so on.
• 📉 Negative exponents represent the reciprocal of the base raised to the corresponding positive exponent, e.g., 10^-1 equals 1/10.
• 🧩 Fractional exponents are calculated by multiplying the numerator by the denominator raised to the power, such as (2/3)^3 equals 8/27.
• 🌱 To convert from exponent to radical form, you can use the formula √(a^n) = b, where b^n = a, with both a and b being positive and n being a natural number.
• 🌐 In adding or subtracting radicals, the radicands (the numbers under the radical sign) must be the same; the radicals themselves are not added or subtracted.

Q & A

What is the definition of 'exponential numbers' as mentioned in the script?

Exponential numbers, also known as 'numbers with exponents', are numbers where a base number is multiplied by itself a certain number of times indicated by the exponent. For example, 3^2 means 3 multiplied by itself 2 times, which equals 9.

How can you quickly calculate the product of exponential numbers with the same base?

You can quickly calculate the product by adding the exponents of the numbers with the same base and then expanding the result. For instance, 3^2 multiplied by 3^3 equals 3^(2+3), which is 3^5, and the result is 243.

What is the rule for dividing exponential numbers with the same base?

The rule for dividing exponential numbers with the same base is to subtract the exponents. For example, 3^3 divided by 3^2 equals 3^(3-2), which is 3^1, and the result is 3.

What happens when you raise a power to another power?

When you raise a power to another power, you multiply the exponents. For example, (3^3) squared means 3 to the power of 3 multiplied by 2, which equals 3^6, and the result is 729.

What is the result of any number raised to the power of zero?

Any number raised to the power of zero equals 1, regardless of the base number. For example, 1^0 equals 1, and 1,000,000^0 also equals 1.

How do you calculate the power of a fraction?

To calculate the power of a fraction, you raise both the numerator and the denominator to the power separately. For example, (2/3)^3 equals (2*2*2)/(3*3*3), which results in 8/27.

How can you convert an exponential number to its root form?

You can convert an exponential number to its root form using the formula √a^n = b, where b is the root of a raised to the power of n. For example, the square root of 25 is 5, as 5^2 equals 25.

What is the condition for adding or subtracting roots?

The condition for adding or subtracting roots is that the radicands (the numbers under the root) must be the same. Only the radicands are added or subtracted, not the roots themselves.

How do you simplify the expression when adding or subtracting square roots that are not prime numbers?

You first try to simplify the radicand to its prime factors or the smallest possible number that can be squared. Then, you can combine the roots if possible. For example, 12√2 minus 6√2 simplifies to 6√2.

What is the scientific notation and why is it important to understand it?

Scientific notation is a standard form used to express very large or very small numbers in the form of a × 10^b, where 1 ≤ a < 10 and b is an integer. It is important to understand because it is globally recognized and commonly used in various scientific fields, including physics.

How do you convert a number like 988,000 into scientific notation?

To convert 988,000 into scientific notation, you move the decimal point 5 places to the left, resulting in 9.88, and then multiply by 10 raised to the power of 5, which gives you 9.88 × 10^5.

Outlines

00:00

📚 Introduction to Exponents and Multiplication Rules

This paragraph introduces the concept of exponents, also known as powers, explaining how a number raised to an exponent means that the base number is multiplied by itself the number of times indicated by the exponent. It provides examples such as 3^2 (3 multiplied by itself twice, resulting in 9) and 2^3 (2 multiplied by itself three times, resulting in 8). The paragraph also discusses a quick method for multiplying numbers with the same base by adding their exponents, as demonstrated with 3^2 multiplied by 3^3, which equals 3^5 or 243. It emphasizes the importance of expanding the terms when the bases are not the same, as in the example of 2^2 multiplied by 3^3, which must be expanded to 2*2*3*3*3*3 and then calculated to get 108.

05:02

🔍 Division and Exponentiation of Exponents

The second paragraph delves into the rules of division for numbers with exponents, which involves subtracting the exponents when the bases are the same, as shown in the example of 3^3 divided by 3^2, which simplifies to 3^1 or 3. It also covers the concept of raising a number with an exponent to another power, which means multiplying the exponents, such as (3^3) squared resulting in 3^6 or 729. The paragraph further explains the process of dealing with exponents in multiplication and division, including the simplification of complex expressions involving square roots and other radicals, and the conversion of exponents to radical form using the formula √(a^n) = b, where b^n = a, with the condition that a and b are positive and n is an integer.

10:03

📉 Exponents with Zero and Negative Bases

This paragraph discusses special cases of exponents, starting with any number raised to the power of zero, which always equals one, regardless of the base, as exemplified by 1^0, 20^0, and a^0 all being equal to 1. It then moves on to negative exponents, which are the reciprocals of the positive exponents, such as 10^-1 being 1/10 and 10^-2 being 1/100. The paragraph also touches on the exponentiation of fractions, where the exponent applies to both the numerator and the denominator, resulting in the multiplication of the fraction by itself the number of times indicated by the exponent, as shown in the example of (2/3)^3 becoming 8/27.

📐 Conversion Between Exponents and Radicals

The final paragraph focuses on converting between exponents and radicals, explaining the process of changing an exponent to its radical form and vice versa, using the formula √(a^n) = b, where b^n = a, with the stipulation that a and b are positive and n is an integer. It provides examples of converting 25 to its square root, which is 5, and then back to its square, which is 25. The paragraph also addresses the simplification of expressions involving addition and subtraction of radicals, emphasizing the need to have the same radicand for addition and subtraction, and the process of simplifying radicals to their prime factors where possible before performing operations.

📘 Rationalizing Radical Denominators and Scientific Notation

The last paragraph discusses the process of rationalizing radical denominators, which involves multiplying the numerator and the denominator by the same radical to eliminate the radical from the denominator. It provides examples of how to simplify expressions with radicals in the denominator and how to handle expressions with both addition and subtraction of radicals. The paragraph also introduces scientific notation as a way to express very large or very small numbers concisely, using the format a × 10^b, where 'a' is a number between 1 and 10, and 'b' is the exponent. It demonstrates how to convert 988,000 into scientific notation, which is 9.88 × 10^5.

Keywords

💡Exponentiation

Exponentiation is a mathematical operation that involves raising a number to the power of another number. In the context of the video, it is the process of multiplying a number by itself a certain number of times, as demonstrated by the example '3^2' which means 3 multiplied by itself twice, resulting in 9. This concept is central to the video's theme of explaining mathematical operations involving powers.

💡Base Number

The base number is the number that is to be raised to a power in an exponentiation operation. In the script, the base number is mentioned in the context of '3' in the expression '3^2', where '3' is the base that is multiplied by itself. Understanding the base number is crucial for grasping how powers and exponents work in mathematics.

💡Power

In mathematics, the power refers to the number of times a base number is multiplied by itself. The script explains that '2' in '2^3' means the base number '2' is multiplied by itself '3' times. This concept is fundamental to the video's educational content on exponents and their calculations.

💡Multiplication of Powers

The script discusses a shortcut for multiplying numbers with the same base by adding their exponents. For example, '3^2 * 3^3' simplifies to '3^5' by adding the exponents 2 and 3. This method is an efficient way to handle multiplication of powers with the same base and is a key point in the video's lesson on exponents.

💡Division of Powers

Division of powers involves reducing the exponent of the base when dividing two powers with the same base. The script uses '3^3 / 3^2' as an example, which simplifies to '3^1' or simply '3', by subtracting the exponents. This operation is part of the video's explanation of how to simplify expressions involving powers.

💡Raising a Power to a Power

When a power is raised to another power, the exponents are multiplied. The video script illustrates this with '(3^3)^2', which simplifies to '3^6' by multiplying the exponents 3 by 2. This concept is an advanced operation on powers discussed in the video.

💡Zero Exponent

The zero exponent rule states that any non-zero base raised to the power of zero equals one. The script mentions '1^0 = 1' and '20^0 = 1' as examples. This rule is an important part of the video's coverage of exponents, showing that the result is always 1 regardless of the base.

💡Negative Exponent

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. The script explains '10^-1' as '1/10' and '10^-2' as '1/100'. This concept is essential for understanding how negative exponents work in mathematical expressions.

💡Fractional Exponent

A fractional exponent represents the nth root of a number. In the script, '(2/3)^3' is an example where the fractional exponent indicates the cube root of the fraction 2/3, resulting in '8/27'. This concept is part of the video's exploration of exponents and roots.

💡Roots

Roots are the inverse operation of exponentiation, where a number is raised to the power of 1/n to find its nth root. The script discusses converting powers to roots and vice versa, using '√25 = 5' and '5^2 = 25' as examples. This concept is integral to the video's explanation of the relationship between roots and powers.

💡Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in the form of a × 10^b, where 1 ≤ |a| < 10, and b is an integer. The script demonstrates converting 988,000 into '9.88 × 10^5'. This notation is a key concept in the video, showing a standardized method for writing large numbers concisely.

Highlights

Introduction to the topic of exponents in 8th-grade mathematics.

Definition of exponents as a way to multiply a number by itself a certain number of times.

Explanation of how to calculate powers of numbers, such as 3^2 meaning 3 multiplied by itself twice.

Quick calculation method for multiplying numbers with the same base by adding their exponents.

Example of multiplying 3^2 by 3^3 resulting in 3^5, demonstrating the exponent addition rule.

Clarification that different base numbers require full expansion before multiplication.

Introduction to the division of exponents, where the exponents are subtracted instead of added.

Example of dividing 3^3 by 3^2, resulting in 3^1, illustrating the exponent subtraction rule.

Explanation of raising a power to another power by multiplying the exponents.

Example of squaring 3^3 to get 3^6 and calculating the result as 729.

Discussion on the rules for exponents when dealing with multiplication within parentheses.

Clarification that any number raised to the power of zero equals one.

Introduction to negative exponents and their interpretation as reciprocals.

Explanation of how to handle exponents with fractions, similar to multiplication rules.

Conversion of exponents to radical form and vice versa using the square root formula.

Example of converting 25 into the square root of 5 and back to its original form.

Guidance on simplifying expressions involving addition and subtraction of radicals.

Process of rationalizing the denominator in radical expressions to avoid radicals in the denominator.

Introduction to scientific notation as a standard way to express large or small numbers.

Example of converting 988,000 into scientific notation as 9.88 x 10^5.

Conclusion and call to action for likes, comments, and subscriptions to the educational channel.

Transcripts

[Musik]

Hai semuanya kembali lagi di channel

portal edukasi Pada kesempatan kali ini

kita akan membahas rangkuman materi

matematika kelas 8 bab 1 yaitu bilangan

berpangkat materi ini sudah kurikulum

Merdeka ya

kita mulai dengan bilangan berpangkat

bilangan berpangkat dikenal juga dengan

istilah bilangan eksponen apabila suatu

angka memiliki pangkat artinya angka

tersebut akan dikalikan dengan angka

yang sama sejumlah nilai pada pangkatnya

contohnya 3 ^ 2 artinya 3 nya ada dua

kali ini dikaliin 3x3 = 9 kemudian 2

pangkat 3 artinya 2 nya dikalikan

sebanyak 3 kali jadinya 2 kali 2 kali 2

= 8 kemudian ada satu pangkat 5 artinya

satunya dikalikan dengan 1 sebanyak 5

kali ya hasilnya 1

selanjutnya perkalian bilangan

berpangkat

pada sistem perkalian bilangan

berpangkat ada cara cepat untuk

menghitung tapi dengan syarat angka

utamanya yaitu sama caranya adalah

dengan menjumlahkan kedua pangkatnya

baru dijabarkan dan dihitung

nih contohnya nih ada tiga pangkat 2

dikali 3 pangkat 3 3 jadinya sama dengan

3 pangkat 2 tambah 3 yaitu = 3 ^ 5

hasilnya 243 bisa dilihat nih angka

utamanya sama yaitu 3 maka untuk

menghitungnya kita tinggal menjumlahkan

pangkatnya saja berbeda apabila angka

utamanya tidak sama maka kita tetap

harus jabarkan satu persatu contohnya 2

^ 2 * 3 ^ 3 nggak bisa nih langsung dua

pangkat 2 ditambah 3 nggak bisa jadinya

dua pangkat duanya dijabarin dua kali

dua tiga pangkat tiganya dijabarin juga

dikali 3 dikali 3 dikali 3 baru dihitung

108

selanjutnya pembagian bilangan

berpangkat

kebalikan dari perkalian Yang pangkat

ini jumlah pada pembagian maka

pangkatnya dikurang contohnya 3 ^ 3

dibagi 3 ^ 2 ini maaf ya bukan kali tapi

dibagi 3 ^ 3 dibagi 3 pangkat 2 jadinya

3 pangkat 3 dikurangi 2 yaitu 3^1 = 3

perbedaan utamanya tetap harus

dijabarkan satu persatu ya

selanjutnya perpangkatan bilangan

berpangkat

apabila suatu angka yang memiliki

pangkat kemudian dipangkatkan kembali

maka pangkatnya dikali contohnya dalam

kurung 3 ^ 3

dikuadratkan artinya 3 pangkat 3 dikali

2 = 3 ^ 6 jadinya Tinggal dihitung nih 3

* 3 * 3 * 3 sebanyak 6 kali = 729

gampang ya

Kemudian perpangkatan pada perkalian

bilangan

Apabila ada dua angka dalam kurung

sedang dikalikan kemudian dipangkatkan

maka cara mengerjakannya bisa dengan

dipaketkan dulu masing-masing baru

dikali contohnya nih dalam kurung 3 * 4

dikuadratkan itu bisa aja jadi sama

dengan 3 pangkat 2 dikali 4 pangkat 2

itu hasilnya 9 kali 16 yaitu 144 ya

walaupun bisa aja sih dikali dulu baru

dipangkatin jangan tanya admin ya kenapa

harus ada cara panjang seperti ini

selanjutnya bilangan pangkat 0

apabila pangkatnya bernilai nol Maka

hasilnya adalah 1 berapapun jumlah angka

utamanya contohnya 1 pangkat 0 = 1 20 ^

0 = 1 1 miliar pangkat 0 = 1 a ^ 0 = 1

bisa dilihat pada contoh di atas

meskipun bukaan k ternyata apabila

dipangkatkan 0 hasilnya adalah 1 Apakah

01 B pangkat 01 ABC pangkat 0 1 semuanya

yang dipake 0 itu adalah 1

selanjutnya bilangan pangkat negatif

kalau ada suatu angka pangkatnya negatif

maka akan menjadi satu persekian

tergantung dari pangkat misalkan 10

pangkat negatif 1 = 1 per 10 pangkat 1

Jadinya 1/10 10 pangkat negatif 2

jadinya 1 per 10 pangkat 2 yaitu = 1 per

100 dan seterusnya

selanjutnya bilangan pecahan berpangkat

jika ada pecahan dipaketkan maka itu

gampang Sistemnya sama seperti perkalian

bilangan perpangkatan

tinggal dikalikan dengan angka yang sama

sejumlah nilai pada pangkatnya misalkan

2/3 ^ 3 jadinya 2/3 * 2/3 * 2/3 jadinya

ya 8/27

selanjutnya mengubah bilangan berpangkat

ke dalam bentuk akar

masih ingat bentuk akar kan Nah sekarang

kita akan mencoba merubah bilangan

berpangkat ke dalam bentuk akar dan juga

sebaliknya kita bisa gunakan rumus di

bawah ini di mana akar dari a

akar dari a ^ n yaitu B = B di mana B

pangkat n = a dengan catatan a dan b

keduanya positif serta n itu bilangan

asli

sebagai contoh nih ada akar pangkat 2

dari 25 yaitu = 5 kita bisa ubah juga 5

^ 2 = 25 bisa kita lihat bahwa nilai

dari n adalah 2 nilai dari a adalah 25

dan nilai b adalah 5

selanjutnya penjumlahan dan pengurangan

bentuk akar

dalam penjumlahan dan pengurangan bentuk

akar ada syarat utama yang harus

dipenuhi yaitu angka dalam akad harus

sama karena yang dijumlahkan adalah

hanya angka utamanya saja akarnya tidak

perlu ditambahkan masih ingat

penjumlahan dan pengurangan aljabar Nah

itu 100% sama ya seperti itu

contohnya 2 akar 5 ditambah 3 akar 5

akar 5 nya nggak perlu ditambahin

jadinya 2 + 3 = 5 √5 kemudian 10 akar 2

dikurangi 8 akar 2 tinggal 10 dikurangi

8 hasilnya 2 9 akar 3 dikurangi 7 √2 Nah

kalau ini nggak bisa ya jadi hasilnya

sama 9 akar 3 dikurangi 7 √2

itu kalau akarnya sudah dalam bilangan

prima atau tidak bisa disederhanakan

lagi akarnya Ya tapi kalau misalkan

seperti ini nih 12 akar 2 dikurangi 3

akar 8 Gimana bisa kita lihat bahwa 8

bukanlah bilangan prima yang artinya

bisa dicoba disederhanakan dulu siapa

tahu bisa dihitung caranya gimana

caranya adalah dengan mengubah bilangan

tersebut menjadi bilangan prima atau

angka yang paling kecil yang paling

mungkin dikalikan sekian yang bisa

memenuhi nilai angka tersebut dimana

angka lainnya bisa disederhanakan dalam

bentuk perpangkatan

langsung contohnya lihat ya nih 3 akar 8

itu bisa kita pecah nih sama dengan 3

akar dari 2 * 4 2 * 4 itu 8 kita bisa

lihat 2 itu bilangan prima dan 4 itu

nanti bisa dipecah nih jadinya 3 dikali

akar 2 dikalikan 4 jadinya 3 dikali akar

2 nih √4 Itu kan bisa disederhanakan

menjadi dua jadinya 3 * √2 * 2 jadinya 6

√2 udah itu baru kita bisa Hitung 12

akar 2 dikurangi 6 √2 jadinya 6√2 Jadi

kalau ngelihat ada soal nih penjumlahan

dan pengurangan kita harus lihat dulu

angkanya udah sekecil mungkin belum nih

yang dalam akar bisa disederhanakan lagi

atau enggak nih jadi kita harus pilih ya

selanjutnya perkalian bentuk akar

ini mirip juga nih dengan perkalian

Aljabar jadi kita kalikan angka utama

dengan angka utama terus dengan akar

kemudian Sederhanakan contohnya 2 akar 3

dikali 2 akar 3

kemudian dikali akar 3 dikali akar 3

jadinya 4 dikali Akar 9 Akar 9 itu kan 3

jadi 4 * 3 = 12 kemudian ada dua akar

dua dikali 2 akar 3 sama nih dua kali

dua dulu kemudian akar 2 dikali akar 3

jadinya 4 dikali akar 6 jadinya 4 akar 6

selanjutnya pembagian bentuk akar

ini juga sama dengan pembagian aljabar

jadi kita bagikan angka utama dengan

angka utama akar dengan akar kemudian

Sederhanakan dengan cara dikali nih

contohnya Ini contoh pertama akar 30 per

akar 3 ingat per itu adalah bagi jadinya

akar 30 per 3 = 30 / 3 10 jadinya akar

10 kemudian contoh kedua yang lebih

sulit dua akar 108/4

√3 2 sama 43

akar dari 108 per akar 3 2/4 kita

Sederhanakan menjadi satu per dua dikali

18 dibagi 3 jadinya 36 akar 36 itu bisa

diselenggarakan juga menjadi 6 jadinya

setengah kali 6 = 3

selanjutnya merasionalkan penyebut

bentuk akar

perlu diingat bahwa bentuk pecahan

penyebutnya tidak boleh dalam bentuk

akar jadi harus dirasionalkan nah cara

merasionalkannya kalau hanya bentuk akar

tunggal maka dikalikan dengan akar angka

akar yang sama per akar angka yang sama

kalau bentuk akarnya Ada ditambah atau

dikurang maka dikalikan dengan angka

yang sama tapi tandanya berbeda Biar

lebih jelas jangan pakai kata-kata deh

ya biarkan angka-angka yang berbicara

contohnya nih contoh pertama

1/√3 ini akarnya tunggal nih cuman √3

doang =

1/√3 *

√3/√3 jadi bisa kita lihat di situ

dikalikannya dengan sama nih √3/√3 = 1 *

√3 1/√3

√3 * √3 jadinya √9 kemudian Akar 9 bisa

disederhanakan nih jadinya = 1 √3/3

kalau ada yang kayak gini kita pecah

satu dengan tiganya jadi 1/3 kemudian

√3-nya dipisah jadi 1/3 √3 contoh kedua

kalau yang nggak tunggal nih ada

pertambahannya

3/2 +

√3 =

3/2 + √3 dikali kan tadi nih tambah nih

2 tambah akar 3 ketika dikalikan terus

dengan lawannya dari tambah menjadi

kurang jadinya 2 dikurangi akar 3 per 2

dikurangi akar 3

kemudian kita kalikan 3 dikali 3 dikali

dalam kurung 2 -√3 dan per 2 tambah akar

3 dikali 2 dikurangi akar 3 3 * 2 yaitu

6 3 dikali negatif akar 3 jadinya

negatif jika akar 3 per ketika kita

sudah hitung ketika ada positif dan

negatif akar 3 nya itu jadi hilang nih

jadinya

2 * 2 4 tinggal

√3 dikali akar 3 jadinya √9 dan tandanya

pasti negatif kalau di situ Jika perlu

dipusingin

kita Sederhanakan 6 dikurangi akar 3 per

akar 3 per Akar 9 sedangkan menjadi 3

jadi 4 dikurangi 3 jadinya sama dengan 6

dikurangi 3 akar 3 per 1 jadi jawabannya

6 dikurangi 3 akar 3

selanjutnya penulisan bentuk baku

penulisan bentuk baku dalam matematika

bisa disebut juga dengan notasi ilmiah

notasi ilmiah adalah bentuk baku dalam

suatu bilangan yang disepakati secara

global kalian wajib memahami notasi

ilmiah ini ya karena akan sangat

digunakan di pelajaran IPA Fisika notasi

ilmiah ini singkatnya meringkas angka 0

atau angka lainnya yang terlalu banyak

menjadi bilangan perpangkatan

biasanya ke dalam bentuk a kali 10

pangkat b dimana nilai a tidak lebih

dari 9,9 dan b adalah seberapa banyak

perpindahannya biar lebih jelas

perhatikan contoh dibawah ini ya biar

paham Ubahlah

988.000 ke dalam notasi ilmiah

Nah kita akan ubah 988.000 ke dalam

notasi ilmiah artinya tidak boleh lebih

dari 9,9 maka menjadi

9,88 yang paling mungkin ya kita hitung

dari paling kanan nih untuk pindah

menjadi angka 9,88 geser Berapa banyak

komanya nih Oh ternyata melewati 5 angka

perhatikan deh dari 0 yang paling

belakang kita jadiin 9,88

12345 baru koma jadinya 9,88 kali 10

pangkat 5 karena selalu dikali 10

pangkat sekian perbedaannya berapa gitu

Nah ya mungkin Cukup sekian terima kasih

telah menyimak video pembelajaran hingga

selesai semoga bermanfaat kita semua

jangan lupa like Comment and subscribe