Artikel Matematika kelas 9 ini menjelaskan tentang bentuk akar dalam matematika, meliputi pengertian, sifat-sifat, dan cara merasionalkannya.
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Apa yang terlintas dalam pikiranmu saat mendengar kata akar? Mungkin kamu membayangkan sebuah pohon yang ditopang oleh akar yang kokoh. Tapi, adakah di antara kamu yang terpikir akar dalam bentuk matematika? Nah, yang akan kita bahas kali ini adalah bentuk akar dalam matematika, ya. Lalu, apa yang dimaksud dengan bentuk akar itu?
Dalam matematika, bentuk akar merupakan suatu operasi aljabar yang dapat digunakan untuk menyelesaikan masalah bilangan. Bentuk akar memiliki sifat-sifat khusus dan dapat dirasionalkan. Apa saja sifat-sifat itu dan bagaimana cara merasionalkan bentuk akar? Simak penjelasan berikut, yuk!
Mengenal Bentuk Akar
Bentuk akar adalah akar dari suatu bilangan rasional yang hasilnya berupa bilangan irasional. Hayo, kamu masih ingat nggak nih dengan bilangan rasional dan irasional? Kalo lupa, bisa baca-baca artikelnya di link ini, ya. Bentuk akar merupakan bentuk lain untuk menyatakan bilangan berpangkat pecahan. Bilangan bentuk akar akan berada dalam tanda “√”, atau bisa kita sebut sebagai tanda akar.
Aku kasih contoh deh biar kamu nggak bingung. Misalnya, ada bilangan berpangkat 21/2. Nah, bilangan berpangkat 21/2 kalo kita ubah ke bentuk akar, jadinya akan seperti ini:
21/2 (a = 2, m = 1, n = 2)
21/2 = atau √2
Fyi nih, kalo indeks akarnya bernilai 2, nggak perlu kamu tulis juga nggak papa, ya.
Contoh bentuk akar yang lain di antaranya √6, √7, √11, dan masih banyak lagi. Coba aku tanya, √25 itu termasuk bentuk akar atau bukan, sih? Eits! Jawabannya bukan bentuk akar. Kenapa? Ingat definisinya, bentuk akar itu berupa bilangan irasional, sedangkan √25 bisa kita sederhanakan menjadi √52 = 52/2 = 5 (5 adalah bilangan rasional). Jadi, √25 bukan bentuk akar. Paham, ya?
Baca Juga: Cara Menyelesaikan Persamaan Kuadrat
Sifat-Sifat Bentuk Akar
Seperti halnya bilangan berpangkat, bilangan bentuk akar juga memiliki sifat-sifat tertentu, lho! Sifat-sifat ini akan memudahkan kita dalam melakukan operasi aljabar yang melibatkan bentuk akar nantinya. Sifat-sifat bentuk akar, di antaranya sebagai berikut:
Nah, setelah kamu mengetahui maksud dari bentuk akar dan sifat-sifatnya, selanjutnya, kita ketahui cara merasionalkan bentuk akar, yuk! Sebeneranya, merasionalkan bentuk akar tuh apa, sih?
Cara Merasionalkan Bentuk Akar
Untuk memudahkan penggunaan bentuk akar dalam operasi aljabar, bentuk akar harus ditulis dalam bentuk yang paling rasional (sederhana). Cara merasionalkan bentuk akar harus memenuhi syarat-syarat tertentu. Syarat-syarat tersebut antara lain sebagai berikut:
Terus, gimana nih kalo misalnya kita menemukan bentuk yang belum sederhana? Gimana cara menyederhanakan bentuk tersebut? Oke, tenang-tenang, aku bakal jelasin caranya di bawah ini.
Kasus 1: Jika bilangan pokok memiliki pangkat lebih besar dari indeks akarnya.
Nah, kalo kamu menemukan bentuk yang kayak gitu, dan bilangan pokoknya itu bernilai positif, maka kamu bisa jabarkan aja bentuk pangkatnya.
Contoh 1: √x5
Bentuk akar √x5 belum sederhana karena pangkat bilangan pokoknya atau pangkat si x lebih besar dari indeks akarnya (5 > 2). Jadi, untuk menyederhanakan bentuk tersebut, kita jabarkan aja pangkat si x nya.
Karena, indeks akarnya itu bernilai 2, maka bisa kita jabarkan kayak gini:
Ingat sifat bentuk akar, ya! Kalo ada operasi perkalian dalam akar, bisa kita pecah jadi seperti ini:
Nah, √x4 itu sama aja dengan x4/2, sehingga bisa disederhanakan menjadi x2. Jadi,
Gimana, paham ya cara menyederhanakannya? Contoh lagi, deh!
Baca Juga: Cara Menyusun Persamaan Kuadrat dan Contohnya
Contoh 2: √20
Kurang lebih cara penyederhanaannya sama kayak contoh 1 kok, teman-teman. Penjabarannya kayak gini,
Itu cara penyederhanaan untuk kasus pertama, ya. Sekarang, kita masuk ke kasus kedua.
Kasus 2: Pada bilangan pecahan, terdapat akar di bagian penyebut.
Kalo kamu menemukan bentuk seperti itu, kamu bisa menyederhanakannya dengan mengalikan bilangan pecahan tersebut dengan bentuk akar yang sekawan dari penyebutnya. Maksudnya bentuk akar yang sekawan tuh gimana, ya? Bentuk akar sekawan itu berarti bentuk akarnya sama, cuma beda tanda operasinya aja. Nah, penjelasan lebih lengkapnya bisa kamu lihat pada gambar di bawah ini!
Biar lebih paham, kita masuk ke contoh soal, ya!
Contoh Soal Bentuk Akar
Contoh: Sederhanakan bentuk akar !
Untuk menyederhanakan bentuk akar tersebut, kita bisa kalikan dengan bentuk akar yang sekawan dari penyebutnya. Karena penyebutnya itu √x, berarti bentuk sekawannya juga √x. Jadi, penyelesaiannya akan seperti ini,
Sudah paham? Kalo gitu, kita masuk ke kasus terakhir.
Kasus 3: Jika di dalam akar memuat bilangan pecahan.
Waduh, gimana nih kalo misalnya kita menemukan soal yang bentuknya kayak gitu? Tenang, kamu masih ingat dengan sifat bentuk akar di atas, kan? Kalo ada pecahan di dalam akar, maka bisa kita jabarkan kayak gini,
Nah, karena setelah dijabarkan bentuknya menjadi seperti kasus nomer 2 (ada akar di penyebut), jadi langkah selanjutnya bisa kita selesaikan seperti kasus nomer 2, teman-teman. Yup! Betul sekali, kita kalikan dengan bentuk akar sekawan penyebutnya. Langsung masuk ke contoh soal aja, deh.
Contoh: Rasionalkan bentuk akar !
Sesuai penjabaran di atas, kita pecah dulu ya bentuk akarnya jadi seperti ini,
Kemudian, kita kalikan dengan bentuk akar sekawan pada penyebutnya. Ingat, pada penyebutnya loh ya, bukan pembilang. Sehingga,
Begitu teman-teman cara merasionalkannya. Sudah paham belum nih sampai sini?
Oke, supaya kamu bisa lebih menguasai materi ini, berikut aku kasih beberapa contoh soal. Bisa kamu kerjakan sendiri atau diskusi dengan teman sekolahmu, ya!
Latihan Soal Bentuk Akar
Sederhanakanlah bentuk akar berikut ini:
Nah, itulah penjelasan mengenai pengertian bentuk akar dalam matematika, sifat-sifat, dan cara merasionalkannya. Jangan lupa untuk terus berlatih soal-soal, ya. Kalo kamu masih ingin mempelajari lagi materi ini, langsung aja gunakan ruangbelajar. Kamu bisa belajar sambil menonton video animasi lengkap dengan soal, pembahasan, dan rangkumannya. Yuk, belajar jadi hebat dengan Ruangguru!
Referensi:
Subchan, Winarni, Hanafi L, dkk. (2015) Matematika SMP/MTs Kelas IX Semester 1. Jakarta: Kementerian Pendidikan dan Kebudayaan
Artikel ini pertama kali dibuat oleh Karina Dwi Adistiana dan diperbarui oleh Hani Ammariah pada 27 Juli 2021.
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<h1 class="content-title">
Ketahui Sifat-Sifat Bentuk Akar & Cara Merasionalkannya | Matematika Kelas 9 </h1>
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<p class="content-author"><a href="https://www.ruangguru.com/blog/author/hani-ammariah" class="">Hani Ammariah</a> </p>
<p class="content-info"><span>September 14, 2022 • </span> <span>5 minutes read</span></p>
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<p><img decoding="async" style="width: 820px; margin-left: auto; margin-right: auto; display: block;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/Bentuk%20Akar%20dan%20Cara%20Merasionalkannya.jpg" alt="Bentuk Akar dan Cara Merasionalkannya" width="820" /></p>
<blockquote>
<p style="font-size: 16px; text-align: center;"><em><a style="font-weight: bold;" href="/blog/tag/matematika-ix" rel="noopener">Artikel Matematika kelas 9</a> ini menjelaskan tentang bentuk akar dalam matematika, meliputi pengertian, sifat-sifat, dan cara merasionalkannya.</em></p>
</blockquote>
<p style="text-align: center;"><em>—</em></p>
<p><span id="more-1624"></span></p>
<p style="text-align: justify;"><span style="color: #3366ff;"><span style="color: #000000;">Apa yang terlintas dalam pikiranmu saat mendengar kata akar? Mungkin kamu membayangkan sebuah pohon yang ditopang oleh akar yang kokoh. Tapi, adakah di antara kamu yang terpikir akar dalam bentuk matematika? <em>Nah</em>, yang akan kita bahas kali ini adalah bentuk akar dalam matematika, ya. Lalu, apa yang dimaksud dengan bentuk akar itu? </span></span></p>
<p style="text-align: justify;"><span style="color: #3366ff;"><span style="color: #000000;">Dalam matematika, bentuk akar merupakan suatu operasi aljabar yang dapat digunakan untuk menyelesaikan masalah bilangan. Bentuk akar memiliki sifat-sifat khusus dan dapat dirasionalkan. Apa saja sifat-sifat itu dan bagaimana cara merasionalkan bentuk akar? Simak penjelasan berikut, <em>yuk</em>!</span></span></p>
<p><span style="color: #3366ff;"><strong> </strong></span></p>
<h2 style="font-size: 20px;">Mengenal Bentuk Akar</h2>
<p style="text-align: justify;"><strong>Bentuk akar adalah akar dari suatu bilangan rasional yang hasilnya berupa bilangan irasional</strong><span style="font-weight: normal;">. Hayo, kamu masih ingat nggak nih dengan bilangan rasional dan irasional? Kalo lupa, bisa baca-baca artikelnya di <a href="/blog/matematika-kelas-7-mengenal-bilangan-rasional-dan-irasional" rel="noopener"><span style="font-weight: bold;">link ini</span></a>, ya. Bentuk akar merupakan bentuk lain untuk menyatakan bilangan berpangkat pecahan. Bilangan bentuk akar akan berada dalam tanda “√”, atau bisa kita sebut sebagai tanda akar. </span></p>
<p style="text-align: justify;"><img decoding="async" style="width: 600px; margin-left: auto; margin-right: auto; display: block;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/Pengertian%20Bentuk%20Akar.jpg" alt="Pengertian Bentuk Akar" width="600" /></p>
<p style="text-align: justify;">
<p style="text-align: justify;">Aku kasih contoh deh biar kamu nggak bingung. Misalnya, ada bilangan berpangkat 2<sup>1/2</sup>. Nah, bilangan berpangkat 2<sup>1/2</sup> kalo kita ubah ke bentuk akar, jadinya akan seperti ini:</p>
<p style="text-align: center;">2<sup>1/2</sup> (a = 2, m = 1, n = 2)</p>
<p style="text-align: center;">2<sup>1/2 </sup>= <img decoding="async" style="width: 27px;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/image-png-Jul-26-2021-07-40-44-21-PM.png" width="27" /> atau √2</p>
<p style="text-align: justify;">Fyi nih, kalo indeks akarnya bernilai 2, nggak perlu kamu tulis juga nggak papa, ya.</p>
<p style="text-align: justify;">Contoh bentuk akar yang lain di antaranya √6, √7, √11, <img decoding="async" style="width: 51px;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/image-png-Jul-26-2021-07-42-00-22-PM.png" width="51" /> dan masih banyak lagi. Coba aku tanya, √25 itu termasuk bentuk akar atau bukan, sih? Eits! Jawabannya bukan bentuk akar. Kenapa? Ingat definisinya, bentuk akar itu berupa bilangan irasional, sedangkan √25 bisa kita sederhanakan menjadi √5<sup>2 </sup>= 5<sup>2/2</sup> = 5 (5 adalah bilangan rasional). Jadi, √25 bukan bentuk akar. Paham, ya?</p>
<p style="text-align: justify; font-weight: bold;">Baca Juga: <a href="/blog/menyelesaikan-persamaan-kuadrat" rel="noopener">Cara Menyelesaikan Persamaan Kuadrat</a></p>
<p style="text-align: justify; font-weight: bold;">
<h2 style="font-size: 20px;">Sifat-Sifat Bentuk Akar</h2>
<p style="text-align: justify;">Seperti halnya <a style="font-weight: bold;" href="/blog/bilangan-berpangkat-pengertian-dan-sifatnya" rel="noopener">bilangan berpangkat</a>, bilangan bentuk akar juga memiliki sifat-sifat tertentu, lho! Sifat-sifat ini akan memudahkan kita dalam melakukan operasi aljabar yang melibatkan bentuk akar nantinya. <span style="font-weight: normal;">Sifat-sifat bentuk akar, di antaranya sebagai berikut</span>:</p>
<p><img decoding="async" style="width: 600px; margin-left: auto; margin-right: auto; display: block;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/Sifat-Sifat%20Bentuk%20Akar%20dan%20Cara%20Merasionalkannya.jpg" alt="Sifat-Sifat Bentuk Akar dan Cara Merasionalkannya" width="600" /></p>
<p> </p>
<p style="text-align: justify;">Nah, setelah kamu mengetahui maksud dari bentuk akar dan sifat-sifatnya, selanjutnya, kita ketahui cara merasionalkan bentuk akar, yuk! Sebeneranya, merasionalkan bentuk akar tuh apa, sih?</p>
<p> </p>
<h2 style="font-size: 20px;">Cara Merasionalkan Bentuk Akar</h2>
<p style="text-align: justify;">Untuk memudahkan penggunaan bentuk akar dalam operasi aljabar, bentuk akar harus ditulis dalam bentuk yang paling rasional (sederhana). <strong>Cara merasionalkan bentuk akar harus memenuhi syarat-syarat tertentu.</strong> Syarat-syarat tersebut antara lain sebagai berikut:</p>
<p><img decoding="async" style="width: 600px; margin-left: auto; margin-right: auto; display: block;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/Cara%20Merasionalkan%20Bentuk%20Akar.jpg" alt="Syarat dalam Merasionalkan Bentuk Akar" width="600" /></p>
<p style="text-align: justify;">
<p style="text-align: justify;">Terus, gimana nih kalo misalnya kita menemukan bentuk yang belum sederhana? Gimana cara menyederhanakan bentuk tersebut? Oke, tenang-tenang, aku bakal jelasin caranya di bawah ini.</p>
<p style="font-weight: bold;">Kasus 1: Jika bilangan pokok memiliki pangkat lebih besar dari indeks akarnya.</p>
<p style="text-align: justify; font-weight: normal;">Nah, kalo kamu menemukan bentuk yang kayak gitu, dan bilangan pokoknya itu bernilai positif, maka kamu bisa jabarkan aja bentuk pangkatnya.</p>
<p style="text-align: justify; font-weight: normal;"><span style="font-weight: bold;">Contoh 1: √x<sup>5</sup></span></p>
<p style="text-align: justify; font-weight: normal;">Bentuk akar √x<sup>5</sup> belum sederhana karena pangkat bilangan pokoknya atau pangkat si x lebih besar dari indeks akarnya (5 > 2). Jadi, untuk menyederhanakan bentuk tersebut, kita jabarkan aja pangkat si x nya.</p>
<p style="text-align: justify; font-weight: normal;">Karena, indeks akarnya itu bernilai 2, maka bisa kita jabarkan kayak gini:</p>
<p style="font-weight: normal; text-align: center;"><img decoding="async" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/image-png-Jul-23-2021-06-34-32-68-AM.png" /></p>
<p style="text-align: justify; font-weight: normal;">Ingat sifat bentuk akar, ya! Kalo ada operasi perkalian dalam akar, bisa kita pecah jadi seperti ini:</p>
<p style="font-weight: normal; text-align: center;"><img decoding="async" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/image-png-Jul-23-2021-06-34-49-34-AM.png" /></p>
<p style="text-align: justify; font-weight: normal;">Nah, √x<sup>4</sup> itu sama aja dengan x<sup>4/2</sup>, sehingga bisa disederhanakan menjadi x<sup>2</sup>. Jadi,</p>
<p style="font-weight: normal; text-align: center;"><img decoding="async" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/image-png-Jul-23-2021-06-35-44-73-AM.png" /></p>
<p style="text-align: justify;">Gimana, paham ya cara menyederhanakannya? Contoh lagi, deh!</p>
<p style="text-align: justify; font-weight: bold;">Baca Juga: <a href="/blog/menyusun-persamaan-kuadrat" rel="noopener">Cara Menyusun Persamaan Kuadrat dan Contohnya</a></p>
<p style="text-align: justify; font-weight: bold;">
<p style="text-align: justify; font-weight: bold;">Contoh 2: √20</p>
<p style="text-align: justify; font-weight: normal;">Kurang lebih cara penyederhanaannya sama kayak contoh 1 kok, teman-teman. Penjabarannya kayak gini,</p>
<p style="text-align: justify; font-weight: normal;"><img decoding="async" style="margin-left: auto; margin-right: auto; display: block;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/image-png-Jul-23-2021-06-43-56-71-AM.png" alt="contoh bentuk akar" /></p>
<p style="text-align: justify; font-weight: normal;">Itu cara penyederhanaan untuk kasus pertama, ya. Sekarang, kita masuk ke kasus kedua.</p>
<p style="font-weight: bold;">Kasus 2: Pada bilangan pecahan, terdapat akar di bagian penyebut.</p>
<p style="text-align: justify;">Kalo kamu menemukan bentuk seperti itu, kamu bisa menyederhanakannya dengan mengalikan bilangan pecahan tersebut dengan bentuk akar yang sekawan dari penyebutnya. Maksudnya bentuk akar yang sekawan tuh gimana, ya? Bentuk akar sekawan itu berarti bentuk akarnya sama, cuma beda tanda operasinya aja. Nah, penjelasan lebih lengkapnya bisa kamu lihat pada gambar di bawah ini!</p>
<p><img decoding="async" style="width: 600px; margin-left: auto; margin-right: auto; display: block;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/MAT09%20-%20Bentuk%20Akar%20dan%20Cara%20Merasionalkannya-06.jpg" alt="Cara Merasionalkan Bentuk Akar" width="600" /></p>
<div></div>
<p>Biar lebih paham, kita masuk ke contoh soal, ya!</p>
<p> </p>
<h2 style="font-size: 20px;">Contoh Soal Bentuk Akar</h2>
<p><span style="font-weight: bold;">Contoh: Sederhanakan bentuk akar <img decoding="async" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/image-png-Jul-26-2021-07-55-59-50-PM.png" />!</span></p>
<p style="text-align: justify;">Untuk menyederhanakan bentuk akar tersebut, kita bisa kalikan dengan bentuk akar yang sekawan dari penyebutnya. Karena penyebutnya itu √x, berarti bentuk sekawannya juga √x. Jadi, penyelesaiannya akan seperti ini,</p>
<p style="text-align: justify;"><img decoding="async" style="width: 182px; margin-left: auto; margin-right: auto; display: block;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/contoh%20soal%20bentuk%20akar.png" alt="contoh soal bentuk akar" width="182" /></p>
<p style="text-align: justify;">Sudah paham? Kalo gitu, kita masuk ke kasus terakhir.</p>
<p style="font-weight: bold;">Kasus 3: Jika di dalam akar memuat bilangan pecahan.</p>
<p style="text-align: justify;">Waduh, gimana nih kalo misalnya kita menemukan soal yang bentuknya kayak gitu? Tenang, kamu masih ingat dengan sifat bentuk akar di atas, kan? Kalo ada pecahan di dalam akar, maka bisa kita jabarkan kayak gini,</p>
<p style="text-align: justify;"><img decoding="async" style="margin-left: auto; margin-right: auto; display: block;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/image-png-Jul-26-2021-08-07-41-85-PM.png" /></p>
<p style="text-align: justify;">Nah, karena setelah dijabarkan bentuknya menjadi seperti kasus nomer 2 (ada akar di penyebut), jadi langkah selanjutnya bisa kita selesaikan seperti kasus nomer 2, teman-teman. Yup! Betul sekali, kita kalikan dengan bentuk akar sekawan penyebutnya. Langsung masuk ke contoh soal aja, deh.</p>
<p style="text-align: justify; font-weight: bold;">Contoh: Rasionalkan bentuk akar <img decoding="async" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/image-png-Jul-26-2021-08-13-54-19-PM.png" />!</p>
<p style="text-align: justify; font-weight: normal;">Sesuai penjabaran di atas, kita pecah dulu ya bentuk akarnya jadi seperti ini,</p>
<p style="font-weight: normal; text-align: center;"><img decoding="async" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/image-png-Jul-26-2021-08-15-35-30-PM.png" /></p>
<p style="font-weight: normal; text-align: justify;">Kemudian, kita kalikan dengan bentuk akar sekawan pada penyebutnya. Ingat, pada penyebutnya loh ya, bukan pembilang. Sehingga,</p>
<p style="font-weight: normal; text-align: justify;"><img decoding="async" style="width: 192px; margin-left: auto; margin-right: auto; display: block;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/contoh%20soal%20bentuk%20akar%201.png" alt="contoh soal bentuk akar" width="192" /></p>
<p style="font-weight: normal; text-align: justify;">Begitu teman-teman cara merasionalkannya. Sudah paham belum nih sampai sini?</p>
<p style="font-weight: normal; text-align: justify;">Oke, supaya kamu bisa lebih menguasai materi ini, berikut aku kasih beberapa contoh soal. Bisa kamu kerjakan sendiri atau diskusi dengan teman sekolahmu, ya!</p>
<p style="font-weight: normal; text-align: justify;">
<h2 style="text-align: justify; font-weight: bold; font-size: 20px;">Latihan Soal Bentuk Akar</h2>
<p style="text-align: justify; font-weight: normal;">Sederhanakanlah bentuk akar berikut ini:</p>
<p style="font-weight: normal; text-align: center;"><img decoding="async" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/image-png-Jul-27-2021-02-39-02-38-AM.png" /></p>
<p style="text-align: justify;">
<p style="text-align: justify;">Nah, itulah penjelasan mengenai pengertian bentuk akar dalam matematika, sifat-sifat, dan cara merasionalkannya. Jangan lupa untuk terus berlatih soal-soal, ya. Kalo kamu masih ingin mempelajari lagi materi ini, langsung aja gunakan <strong><a href="https://ruangguru.com/belajar?utm_medium=referral&utm_source=blog&utm_campaign=mengenal-bilangan-bentuk-akar" target="_blank" rel="noopener">ruangbelajar</a></strong>. Kamu bisa belajar sambil menonton video animasi lengkap dengan soal, pembahasan, dan rangkumannya. Yuk, <span style="font-weight: normal;">belajar jadi hebat</span> dengan Ruangguru!</p>
<p style="text-align: center;"><a class="rg-cta" style="text-align: center;" href="https://link.ruangguru.com/blPk/eaff0eb9" target="_blank" rel="noopener"><img fetchpriority="high" decoding="async" src="https://cdn-web.ruangguru.com/landing-pages/assets/cta/0ec4a2df-db2e-4ae1-8473-692756dcb684.jpeg" alt="IDN CTA Blog ruangbelajar Ruangguru" width="822" height="200" /></a></p>
<p style="text-align: left;"><strong>Referensi:</strong></p>
<p style="text-align: left;">Subchan, Winarni, Hanafi L, dkk. (2015) Matematika SMP/MTs Kelas IX Semester 1. Jakarta: Kementerian Pendidikan dan Kebudayaan</p>
<p style="text-align: left;">
<p style="text-align: left;"><em>Artikel ini pertama kali dibuat oleh Karina Dwi Adistiana dan diperbarui oleh Hani Ammariah pada 27 Juli 2021.</em></p>
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<h5 class="author-title">Hani Ammariah</h5>
<p class="author-description">Content Performance Coordinator at Ruangguru. I like to write things that are useful and also not. But I hope, you can learn something from this post. ^^</p>
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<div class="d-lg-none text-center">
<nav class="footer-menu social">
<p class="title">Ikuti Kami</p>
<a target="_blank" href="https://www.instagram.com/ruangguru/">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/instagram.png?height=60&convert=webp" type="image/webp">
<source srcset="https://cdn-web-2.ruangguru.com/static/instagram.png?height=60" type="image/png">
<img src="https://cdn-web-2.ruangguru.com/static/instagram.png" loading="lazy" alt="Logo Instagram Footer" width="32px" height="32px" />
</picture>
</a>
<a target="_blank" href="https://www.facebook.com/ruanggurucom/">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/facebook.png?height=60&convert=webp" type="image/webp">
<source srcset="https://cdn-web-2.ruangguru.com/static/facebook.png?height=60" type="image/png">
<img src="https://cdn-web-2.ruangguru.com/static/facebook.png" loading="lazy" alt="Logo Facebook Footer" width="32px" height="32px" />
</picture>
</a>
<a target="_blank" href="https://twitter.com/ruangguru/">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/twitter.png?height=60&convert=webp" type="image/webp">
<source srcset="https://cdn-web-2.ruangguru.com/static/twitter.png?height=60" type="image/png">
<img src="https://cdn-web-2.ruangguru.com/static/twitter.png" loading=" lazy" alt="Logo Twitter Footer" width="32px" height="32px" />
</picture>
</a>
<a target="_blank" href="https://www.youtube.com/ruangguru">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/youtube.png?height=60&convert=webp" type="image/webp">
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<img src="https://cdn-web-2.ruangguru.com/static/youtube.png" loading=" lazy" alt="Logo Youtube Footer" width="32px" height="32px" />
</picture>
</a>
<a target="_blank" href="https://www.linkedin.com/company/ruangguru-com">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/linkedin.png?height=60&convert=webp" type="image/webp">
<source srcset="https://cdn-web-2.ruangguru.com/static/linkedin.png?height=60" type="image/png">
<img src="https://cdn-web-2.ruangguru.com/static/linkedin.png" loading=" lazy" alt="Logo LinkedIn Footer" width="32px" height="32px" />
</picture>
</a>
<a target="_blank" href="https://www.tiktok.com/@ruangguru">
<picture>
<source srcset="https://cdn-web-2.ruangguru.com/static/tiktok.png?height=60&convert=webp" type="image/webp">
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<img src="https://cdn-web-2.ruangguru.com/static/tiktok.png" loading=" lazy" alt="Logo TikTok Footer" width="32px" height="32px" />
</picture>
</a>
</nav>
</div>
<p class="footer-copyright">
© 2026 All Rights Reserved PT. Ruang Raya Indonesia
</p>
</div>
<style>
.bantuan-container {
position: relative;
}
.bantuan-element {
position: fixed;
z-index: 999;
display: flex;
align-items: flex-end;
justify-content: flex-end;
cursor: pointer;
transition: all 0.6s ease;
}
.bantuan-top {
bottom: 11rem;
right: 2rem;
}
.bantuan-bottom {
bottom: 2rem;
right: 2rem;
}
.bantuan-top-2 {
bottom: 15rem;
right: 2rem;
}
.bantuan-top-3 {
bottom: 19rem;
right: 2rem;
}
.bantuan-bottom-2 {
bottom: 5rem;
right: 2rem;
}
.bantuan-bottom-3 {
bottom: 8rem;
right: 2rem;
}
@media (min-width: 768px) {
.bantuan-element img {
width: auto;
transition: all 0.4s ease;
}
.bantuan-top img {
width: 140px;
}
.bantuan-bottom img {
width: 140px;
}
.bantuan-top-2 img,
.bantuan-top-3 img {
width: 140px;
}
.bantuan-bottom-2 img,
.bantuan-bottom-3 img {
width: 140px;
}
.bantuan-element.clicked {
bottom: 9rem;
right: 2.5rem;
}
.bantuan-element.clicked img {
width: auto;
height: 64px;
}
}
@media (max-width: 767px) {
.bantuan-top {
bottom: 8rem;
right: 1rem;
align-items: center;
}
.bantuan-bottom {
bottom: 2rem;
right: 1.25rem;
align-items: center;
}
.bantuan-top-2 {
bottom: 12rem;
right: 1rem;
align-items: center;
}
.bantuan-top-3 {
bottom: 16rem;
right: 1rem;
align-items: center;
}
.bantuan-bottom-2 {
bottom: 5rem;
right: 1.25rem;
align-items: center;
}
.bantuan-bottom-3 {
bottom: 8rem;
right: 1.25rem;
align-items: center;
}
.bantuan-element img {
width: auto;
}
.bantuan-top img,
.bantuan-top-2 img,
.bantuan-top-3 img {
width: 96px;
}
.bantuan-bottom img,
.bantuan-bottom-2 img,
.bantuan-bottom-3 img {
width: 96px;
}
}
.bantuan-element:hover {
transform: scale(1.05);
}
.bantuan-element img {
max-width: 100%;
height: auto;
}
</style>
<div id="bantuan-divs-container" style="position: relative" data-product="ruangguru" data-table="rg_floating_icon"></div>
<script>
class FloatingDivsComponent {
constructor(product = 'ruangguru', table = 'rg_floating_icon') {
this.container = document.getElementById('bantuan-divs-container');
this.floatingElements = [];
this.product = product;
this.table = table;
// Configuration
this.config = {
baseApiUrl: 'https://gw.ruangguru.com/api/v3/rdb/p_ddgy8hjx5dxsqx/rg_banner/views/',
apiKey: 'vLyK3dJrDBRmOwueX2OJXibMkgsUbBk0WS4ua2p1',
maxElements: 2,
zIndexBase: 100,
};
// Build dynamic API URL based on table parameter
this.config.apiUrl = this.buildApiUrl();
this.init();
}
buildApiUrl() {
const baseUrl = this.config.baseApiUrl;
const tableName = this.table || 'rg_floating_icon';
const queryParams =
'offset=0&limit=99999&sort=rankPosition&where=(type,eq,floating)~and(showIn,anyof,blog)~and(hide,is,false)~and(product,eq,' +
this.product +
')&nested%5Bcta%5D%5Bfields%5D=Id,name,ctaId,imageUrl,ctaLink,targetUrl';
return `${baseUrl}${tableName}?${queryParams}`;
}
async init() {
try {
await this.fetchFloatingElements();
this.renderFloatingElements();
} catch (error) {
console.error('Failed to initialize floating divs component:', error);
}
}
async fetchFloatingElements() {
try {
const options = {
method: 'GET',
headers: {
'xc-token': this.config.apiKey || 'vLyK3dJrDBRmOwueX2OJXibMkgsUbBk0WS4ua2p1',
},
};
const urlNoco = this.config.apiUrl;
const response = await fetch(urlNoco, options);
if (!response.ok) {
throw new Error(`HTTP error! status: ${response.status}`);
}
const data = await response.json();
this.floatingElements = this.processApiResponse(data);
} catch (error) {
console.error('Error fetching floating elements:', error);
throw error;
}
}
processApiResponse(data) {
if (!data || !data.list || !Array.isArray(data.list)) {
return [];
}
console.log(data.list);
// Ensure we only get maximum 2 elements
const limitedElements = data.list.slice(0, this.config.maxElements);
return limitedElements;
}
renderFloatingElements() {
if (!this.container || this.floatingElements.length === 0) {
return;
}
// Additional safeguard: ensure we never render more than maxElements
const elementsToRender = this.floatingElements.slice(0, this.config.maxElements);
// Clean up existing elements and their event listeners
this.cleanupExistingElements();
// Clear existing content
this.container.innerHTML = '';
elementsToRender.forEach((element, index) => {
const floatingDiv = this.createFloatingDiv(element, index);
this.container.appendChild(floatingDiv);
});
}
cleanupExistingElements() {
// Clear existing elements (no need for event listener cleanup with CSS approach)
const existingElements = this.container.querySelectorAll('[data-floating-element]');
existingElements.forEach(element => {
element.remove();
});
}
createFloatingDiv(element, index) {
const div = document.createElement('div');
const cta = element.cta[0]; // Use the first CTA
// Add data attribute for cleanup tracking
div.setAttribute('data-floating-element', 'true');
// Apply CSS classes based on rankPosition
const positionClass = this.getPositionClass(element.rankPosition, index);
div.className = `bantuan-element ${positionClass}`;
div.style.zIndex = this.config.zIndexBase + index;
// Create the link and image
const link = document.createElement('a');
link.href = 'https://cta.ruangguru.com/' + cta.ctaId || '#';
link.target = '_blank';
// Add tracking attributes
link.setAttribute('data-tracking-event-type', 'landing_page__core__item_click');
link.setAttribute(
'data-tracking-context',
JSON.stringify({
product_name: this.product,
cta_action: 'redirection',
cta_action_detail: link.href,
item_type: 'floating',
item_name: element.name || 'Floating element',
item_category: 'payment',
item_position: index,
item_serial: null,
section_type: 'floating_item',
section_position: 'body',
section_name: 'floating_item',
})
);
const img = document.createElement('img');
img.src = element.imageUrl + '?h=200&convert=webp';
img.alt = element.name || 'Floating element';
img.loading = 'lazy';
// Add click tracking
link.addEventListener('click', () => {
this.trackClick(element, cta);
});
link.appendChild(img);
div.appendChild(link);
return div;
}
getPositionClass(rankPosition, index) {
// Define positioning classes based on rankPosition
if (rankPosition === 1) {
return 'bantuan-top'; // Top position
} else if (rankPosition === 2) {
return 'bantuan-bottom'; // Bottom position
} else {
// For additional elements, distribute them evenly
return `bantuan-${index % 2 === 0 ? 'top' : 'bottom'}-${Math.floor(index / 2) + 1}`;
}
}
trackClick(element, cta) {
// Implement click tracking logic here
console.log('Floating element clicked:', {
elementId: element.Id,
elementName: cta.name,
ctaId: cta.ctaId,
targetUrl: cta.targetUrl,
});
}
// Public method to manually refresh
async refresh() {
try {
await this.fetchFloatingElements();
this.renderFloatingElements();
} catch (error) {
console.error('Failed to refresh floating elements:', error);
throw error;
}
}
// Public method to get current floating elements
getFloatingElements() {
return [...this.floatingElements];
}
}
// Initialize the component when DOM is ready
document.addEventListener('DOMContentLoaded', () => {
// Get product and table parameters from data attributes
const container = document.getElementById('bantuan-divs-container');
const product = container.getAttribute('data-product') || 'ruangguru';
const table = container.getAttribute('data-table') || 'rg_floating_icon';
new FloatingDivsComponent(product, table);
});
// Export for use in other modules if needed
if (typeof module !== 'undefined' && module.exports) {
module.exports = FloatingDivsComponent;
}
</script> <div id="wp-install-nudge" style="
position: fixed;
bottom: 0rem;
right: 0;
z-index: 999;
cursor: pointer;
"></div>
<script>
(function (w, d, s, o, f, js, fjs) {
w['Sales-Nudge-Widget'] = o; w[o] = w[o] || function () { (w[o].q = w[o].q || []).push(arguments) };
js = d.createElement(s), fjs = d.getElementsByTagName(s)[0];
js.id = o; js.src = f; js.async = 1; fjs.parentNode.insertBefore(js, fjs);
}(window, document, 'script', 'wpInstallNudge', 'https://cdn-web-2.ruangguru.com/sales-nudge/6198a95130/dist/widget.js'));
wpInstallNudge('init', { targetElementId: 'wp-install-nudge', clientTarget: 'ruangguru_blog' });
</script>
<script>
// note: disabled due to auto reload page
// var lastScrollTop = 0;
// $(window).scroll(function(event){
// var st = $(this).scrollTop();
// if (st > lastScrollTop){
// $('#wp-install-nudge').slideDown( "slow", function() {});
// } else {
// $('#wp-install-nudge').slideUp( "slow", function() {});
// }
// lastScrollTop = st;
// });
// $(document).ready(function () {
// $("#wp-install-nudge").hide();
// });
$('#wp-install-nudge').click(function(){
Analytics.track("promotion__install_nudge__click",
{
user_email: "ruangguru@blog.com",
cta_link: widgetData.dataConfig.target_link
}).catch((err) => {
console.log("tracker error", err);
});
});
</script>
</footer>
<div id="__next"></div>
<script type="speculationrules">
{"prefetch":[{"source":"document","where":{"and":[{"href_matches":"\/blog\/*"},{"not":{"href_matches":["\/blog\/wp-*.php","\/blog\/wp-admin\/*","\/blog\/wp-content\/uploads\/*","\/blog\/wp-content\/*","\/blog\/wp-content\/plugins\/*","\/blog\/wp-content\/themes\/ruangguru\/*","\/blog\/*\\?(.+)"]}},{"not":{"selector_matches":"a[rel~=\"nofollow\"]"}},{"not":{"selector_matches":".no-prefetch, .no-prefetch a"}}]},"eagerness":"conservative"}]}
</script>
<script>
async function getCtaDetails(ctaIds) {
const endpoint =
" https://gw.ruangguru.com/api/v3/rdb/p_ddgy8hjx5dxsqx/CTA/views/CTA?offset=0&limit=10&where=(ctaId,in," + ctaIds.toString() + ")";
try {
const response = await fetch(endpoint, {
headers: {
"xc-token": "-hG2nBDJzSj0agSgNtcJx97M4F02y54nUMoY6plJ",
},
});
const data = await response.json();
const ctaList = data?.list || [];
const ctaMap = new Map();
for (const ctaData of ctaList) {
ctaMap.set(ctaData.ctaId, ctaData);
}
return ctaMap;
} catch (error) {
console.error("Error fetching CTA details:", error);
return new Map();
}
}
async function replaceCTA() {
const regex = /{{cta\((‘(.*?)’|“(.*?)”)\)}}/g;
const bodyHTML = document.body.innerHTML;
const matches = Array.from(bodyHTML.matchAll(regex));
const ctaIds = matches.map((match) => {
let ctaId = match[2] ?? match[3];
return ctaId;
});
const ctaMap = await getCtaDetails(ctaIds);
const replacedHTML = bodyHTML.replace(regex, (match, _inner, ctaId1, ctaId2) => {
const ctaId = ctaId1 ?? ctaId2;
const ctaData = ctaMap.get(ctaId);
if (ctaData) {
return `<a href="https://cta.ruangguru.com/${ctaData.ctaId}" target="_${ctaData.targetType}" style="position:relative;display:inline-block;">
<img src="${ctaData.imageUrl}" loading="lazy" alt="${ctaData.altText}" />
</a>`;
} else {
return match;
}
});
document.body.innerHTML = replacedHTML;
}
</script>
<meta name="author" content="Ruangguru Tech Team">
<!-- Google Tag Manager (noscript) -->
<noscript><iframe src="https://www.googletagmanager.com/ns.html?id=GTM-PVNLZFT" height="0" width="0" style="display:none;visibility:hidden"></iframe></noscript>
<!-- Tracker -->
<script type="text/javascript" src="https://cdn-web-2.ruangguru.com/landing-page-web/public/staticpages/global-assets/js/cta.js"></script>
<script src="https://unpkg.com/@dotlottie/player-component@latest/dist/dotlottie-player.mjs" type="module"></script>
</body>
</html>