Pada artikel Matematika kelas 7 kali ini, kamu akan mengetahui cara menyelesaikan operasi perpangkatan bentuk aljabar. Yuk, kita pelajari selengkapnya!
—
Jika pada artikel sebelumnya kamu telah mengetahui tentang bentuk aljabar dan cara menyelesaikan beberapa operasi hitung aljabar, maka pada artikel kali ini, kita akan lanjut membahas operasi hitung aljabar yang lainnya, yaitu operasi perpangkatan. Tapi sebelum itu, ayo kita mengingat kembali apa itu aljabar, ya.
Hayo, siapa di antara kamu yang masih ingat apa itu aljabar?
Aljabar adalah salah satu dari cabang Matematika yang mempelajari tentang penyelesaian masalah menggunakan huruf-huruf untuk mewakili angka-angka.
Bentuk aljabar terdiri dari variabel (nilai berubah), konstanta (nilai tetap), dan koefisien (faktor pengali). Misalnya, 2a + 1 nih. Berarti, 2 merupakan koefisiennya, a merupakan variabelnya, dan 1 merupakan konstantanya.
Bagaimana, sampai di sini kamu sudah ingat?
Oke, kalau begitu, selanjutnya mari kita masuk ke operasi perpangkatan pada aljabar, ya. Kira-kira, bagaimana sih cara untuk menyelesaikan operasi perpangkatan pada aljabar? Yuk, langsung saja kita simak pada artikel di bawah ini. Let’s scroll it!
Konsep Perpangkatan Bentuk Aljabar
Sebelumnya, ayo kita simak kisah Rogu berikut ini dan kita selesaikan bersama-sama, ya!
Pada saat upacara bendera di sekolah, Rogu bergabung ke dalam grup paduan suara yang selama ini ia idam-idamkan. Grup tersebut terdiri dari 5 baris.
Baris pertama, bertugas untuk memainkan pianika, baris kedua bertugas untuk memainkan recorder, dan baris ketiga sampai kelima bertugas untuk menyanyikan lagu Indonesia Raya dan Mengheningkan Cipta.
Rogu, berada pada baris pertama yang bertugas untuk memainkan pianika. Setiap baris terdiri dari 5 orang anak. Dapatkah kamu menghitung berapa jumlah seluruh anak yang bergabung dalam grup paduan suara tersebut?
Baca Juga: Pengertian, Jenis, dan Contoh Bilangan Bulat
Nah, untuk menghitung jumlah seluruh anak dalam grup tersebut, kamu dapat menggunakan cara perkalian sebagai berikut:
5 × 5
Ternyata, perkalian di atas termasuk salah satu contoh dari perkalian berulang, lho. Kenapa? Karena perkalian tersebut terdiri dari bilangan dengan faktor-faktor yang sama, yaitu 5.
Tahukah kamu, setiap perkalian berulang dapat ditulis secara ringkas dengan menggunakan notasi bilangan berpangkat seperti di bawah ini:
5² (dibaca 5 pangkat 2)
Jadi jawabannya sudah pada tahu ya, yaitu 5² = 5 × 5 = 25 orang anak.
Oke, agar kamu lebih paham lagi tentang notasi bilangan berpangkat, yuk perhatikan contoh berikut:
2 × 2 × 2 = 2³ (dibaca 2 pangkat 3)
3 × 3 × 3 × 3 = 34 (dibaca 3 pangkat 4)
a × a × a × … × a = an (dibaca a pangkat n)
Jika kamu melihat contoh-contoh di atas, maka dapat kamu ketahui kalau perpangkatan adalah suatu bilangan yang dikalikan dengan dirinya sendiri sebanyak jumlah pangkatnya (n kali).
Pada prinsipnya, ternyata rumus perpangkatan pada bentuk aljabar sama dengan perpangkatan pada bilangan bulat.
Rumus Perpangkatan pada Bentuk Aljabar
Bentuk Istimewa dalam Perpangkatan Bentuk Aljabar
Selain itu, terdapat beberapa bentuk istimewa yang akan sering kamu temui dalam perpangkatan aljabar, yaitu:
Contoh Soal Perpangkatan Bentuk Aljabar
Sekarang, yuk, coba kerjakan soal di bawah ini supaya kamu semakin paham!
Contoh 1
Penyelesaian:
Baca Juga: Mengenal Operasi Hitung pada Pecahan
Contoh 2
Penyelesaian:
Nah, untuk contoh yang satu ini, penyelesaiannya cukup rumit nih. Jadi, simak baik-baik ya langkah-langkahnya. Oke, langkah pertama yang bisa kamu lakukan untuk menyelesaikan soal di atas adalah operasikan persamaan tersebut seperti biasa.
Kemudian, langkah keduanya adalah kita samakan variabelnya.
Berdasarkan persamaan (3), didapat nilai q = 4. Lalu, kita substitusikan nilai q ke persamaan (2) menjadi sebagai berikut:
Setelah kita substitusikan niai q ke persamaan (2), didapat nilai p = 5. Selanjutnya, kita substitusikan nilai p ke persamaan (1) menjadi sebagai berikut:
Sehingga, diperoleh nilai r adalah 10.
Bagaimana sampai di sini? Paham atau paham? Bagi kamu yang masih kurang paham, jangan ragu untuk tuliskan pertanyaanmu di kolom komentar, ya.
Berdasarkan penjabaran di atas, sekarang kamu sudah tahu bagaimana cara menyelesaikan operasi perpangkatan pada bentuk aljabar berpangkat dua, kan.
Lalu, bagaimana cara untuk menyelesaikan operasi perpangkatan pada bentuk aljabar yang pangkatnya lebih dari dua? Jawabannya ada dua cara, nih. Penasaran bagaimana caranya? Kalau begitu, yuk langsung simak contohnya di bawah ini!
Penyelesaian:
- Cara 1: Menggunakan pola segitiga pascal
Sebelumnya, ayo kita perhatikan ilustrasi gambar pola segitiga pascal berikut ini dulu, ya!
Segitiga pascal (sumber:en.wikipedia.org)
Pada gambar di atas dapat kamu ketahui bahwa, pola bilangan tersebut membentuk bangun segitiga yang selalu diawali dan diakhiri dengan angka 1.
Kemudian, bilangan-bilangan yang selain angka 1 itu diperoleh dari jumlah dua buah bilangan yang terletak di atasnya dan saling berdekatan.
Pola segitiga pascal ini ternyata berhubungan dengan koefisien pada bentuk aljabar, lho. Coba kamu perhatikan gambar di bawah ini!
Ternyata, koefisien pada bentuk aljabar di atas sama dengan pola segitiga pascal. Selanjutnya, coba kamu perhatikan besar pangkat pada masing-masing variabelnya.
Ternyata, semakin ke kanan, besar pangkat variabel a akan semakin kecil (an –> an-1 –> … –> a0) dan besar pangkat variabel b akan semakin besar (b0 –> … –> bn).
Jadi, berdasarkan soal di atas dapat kita peroleh hasil sebagai berikut:
Paham ya dengan cara segitiga pascal ini. Oke, kalau begitu kita lanjut ke cara yang kedua, ya.
Baca Juga: Jenis-Jenis Bilangan Pecahan dan Contohnya
Nah, bagi kamu yang malas menghafal pola segitiga pascal di atas, kamu bisa menggunakan cara yang satu ini. Pada cara ini, kamu hanya tinggal menguraikan aljabar tersebut menjadi sebagai berikut:
Wah, lumayan panjang ya. Mungkin, untuk menyelesaikan bentuk aljabar dengan pangkat 3 dan 4, cara ini masih cukup efektif. Tapi, bagaimana dengan bentuk aljabar yang pangkatnya 5, 6, 7, dan seterusnya?
Jangan sampai waktu kamu habis hanya untuk menyelesaikan bentuk aljabar yang seperti itu, ya.
Oleh karena itu, ada pentingnya juga nih untuk selalu mengingat pola segitiga pascal yang telah kita pelajari sebelumnya. Selain itu, dalam mengerjakan soal-soal aljabar ini juga dibutuhkan ketelitian yang tinggi.
Perhatikan selalu tanda (+) dan (-) nya karena kedua tanda itu akan berubah ketika kamu melakukan operasi perkalian dan pemangkatan.
—
So, bagaimana tanggapanmu setelah membaca artikel ini? Mudah atau sulit?
Perlu kita akui kalau materi aljabar ini memang cukup rumit, ya. Oleh karena itu, kamu juga perlu untuk memperbanyak latihan soal, nih.
Nah, bagi kamu yang masih belum paham dengan materi ini dan ingin tanya-tanya lebih lanjut lagi, kamu bisa lho gabung dengan Ruangguru Privat. Belajar jadi semakin asik karena kamu bisa berdiskusi dengan kakak-kakak Tutor lulusan PTN terbaik di Indonesia!
Referensi:
Manik D.R, (2009) Penunjang Belajar Matematika untuk SMP/MTs Kelas 7. Jakarta: Pusat Perbukuan, Departemen Pendidikan Nasional.
Sumber Foto:
Ilustrasi ‘Segitiga Pascal’ [Daring]. Tautan: https://upload.wikimedia.org/wikipedia/commons/0/0d/PascalTriangleAnimated2.gif (Diakses: 28 Desember 2020)
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<h1 class="content-title">
Cara Menyelesaikan Operasi Perpangkatan pada Bentuk Aljabar | Matematika Kelas 7 </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>July 8, 2025 • </span> <span>6 minutes read</span></p>
<br />
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<p><img decoding="async" class="aligncenter size-medium" src="https://cdn-web-2.ruangguru.com/landing-pages/assets/e57bd776-048b-4888-a1d1-9644a4f009dd.png" alt="cara menyelesaikan operasi perpangkatan bentuk aljabar" /></p>
<blockquote>
<p style="text-align: center;"><em>Pada <strong><a href="https://www.ruangguru.com/blog/c/matematika/matematika-smp-kelas-7">artikel Matematika kelas 7</a></strong> kali ini, kamu akan mengetahui <strong>cara menyelesaikan operasi perpangkatan bentuk aljabar</strong>. Yuk, kita pelajari selengkapnya!</em></p>
</blockquote>
<p style="text-align: center;"><em>—</em></p>
<p> </p>
<p style="text-align: justify;"><span style="background-color: transparent;">Jika pada artikel sebelumnya kamu telah mengetahui tentang </span><strong><a href="https://www.ruangguru.com/blog/penyelesaian-bentuk-bentuk-aljabar" target="_blank" rel="noopener">bentuk aljabar dan cara menyelesaikan beberapa operasi hitung aljabar</a></strong><span style="background-color: transparent;">, maka pada artikel kali ini, kita akan lanjut membahas operasi hitung aljabar yang lainnya, yaitu operasi perpangkatan. Tapi sebelum itu, ayo kita mengingat kembali apa itu aljabar, ya.</span><span style="background-color: transparent;"> </span></p>
<p style="text-align: justify;">Hayo, siapa di antara kamu yang masih ingat apa itu aljabar?</p>
<p style="text-align: justify;"><strong>Aljabar adalah </strong>salah satu dari cabang Matematika yang mempelajari tentang<strong> penyelesaian masalah menggunakan huruf-huruf untuk mewakili angka-angka. </strong></p>
<p style="text-align: justify;">Bentuk aljabar terdiri dari <strong>variabel (nilai berubah), konstanta (nilai tetap), dan koefisien (faktor pengali)</strong>. Misalnya, 2a + 1 nih. Berarti, 2 merupakan koefisiennya, a merupakan variabelnya, dan 1 merupakan konstantanya.</p>
<p style="text-align: justify;">Bagaimana, sampai di sini kamu sudah ingat?</p>
<p style="text-align: justify;">Oke, kalau begitu, selanjutnya mari kita masuk ke operasi perpangkatan pada aljabar, ya. Kira-kira, bagaimana sih cara untuk menyelesaikan operasi perpangkatan pada aljabar? Yuk, langsung saja kita simak pada artikel di bawah ini. <em>Let’s scroll it</em>!</p>
<p> </p>
<h2><span style="font-size: 18pt;">Konsep Perpangkatan Bentuk Aljabar</span></h2>
<p style="text-align: justify;">Sebelumnya, ayo kita simak kisah Rogu berikut ini dan kita selesaikan bersama-sama, ya!</p>
<p><img decoding="async" style="width: 600px; display: block; margin: 0px auto;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/soal%20operasi%20perhitungan%20bentuk%20aljabar.png" alt="soal operasi perhitungan bentuk aljabar" width="600" /></p>
<p> </p>
<p style="text-align: justify;">Pada saat upacara bendera di sekolah, Rogu bergabung ke dalam grup paduan suara yang selama ini ia idam-idamkan. Grup tersebut terdiri dari 5 baris.</p>
<p style="text-align: justify;">Baris pertama, bertugas untuk memainkan pianika, baris kedua bertugas untuk memainkan <em>recorder</em>, dan baris ketiga sampai kelima bertugas untuk menyanyikan lagu Indonesia Raya dan Mengheningkan Cipta.</p>
<p style="text-align: justify;">Rogu, berada pada baris pertama yang bertugas untuk memainkan pianika. Setiap baris terdiri dari 5 orang anak. Dapatkah kamu menghitung berapa jumlah seluruh anak yang bergabung dalam grup paduan suara tersebut?</p>
<p><strong>Baca Juga: <a href="https://www.ruangguru.com/blog/pengertian-dan-contoh-bilangan-bulat">Pengertian, Jenis, dan Contoh Bilangan Bulat</a></strong></p>
<p style="text-align: justify;">Nah, untuk menghitung jumlah seluruh anak dalam grup tersebut, kamu dapat menggunakan cara perkalian sebagai berikut:</p>
<p style="text-align: center;"><span style="font-size: 24pt;">5 × 5</span></p>
<p> </p>
<p style="text-align: justify;">Ternyata, perkalian di atas termasuk salah satu contoh dari <strong>perkalian berulang</strong>, <em>lho</em>. Kenapa? Karena <strong>perkalian tersebut terdiri dari bilangan dengan faktor-faktor yang sama</strong>, yaitu 5.</p>
<p style="text-align: justify;">Tahukah kamu, setiap <strong>perkalian berulang dapat ditulis secara ringkas dengan menggunakan notasi bilangan berpangkat</strong> seperti di bawah ini:</p>
<p style="text-align: center;"><span style="font-size: 30px;">5²</span><span style="text-align: center; background-color: transparent;"> (dibaca 5 pangkat 2)</span></p>
<p> </p>
<p>Jadi jawabannya sudah pada tahu ya, yaitu 5² = 5 <span style="font-size: 14pt;">×</span> 5 = 25 orang anak.</p>
<p>Oke, agar kamu lebih paham lagi tentang notasi bilangan berpangkat, <em>yuk</em> perhatikan contoh berikut:</p>
<p style="text-align: center;"><span style="font-size: 14pt;">2 × 2 × 2 = 2³</span> (dibaca 2 pangkat 3)</p>
<p style="text-align: center;"><span style="font-size: 14pt;">3 × 3 × 3 × 3 = 3<sup>4</sup></span> (dibaca 3 pangkat 4)</p>
<p style="text-align: center;"><span style="font-size: 14pt;">a × a × a × … × a = a<sup>n</sup></span> (dibaca a pangkat n)</p>
<p> </p>
<p style="text-align: justify;">Jika kamu melihat contoh-contoh di atas, maka dapat kamu ketahui kalau <strong>perpangkatan adalah suatu bilangan yang dikalikan dengan dirinya sendiri sebanyak jumlah pangkatnya (n kali)</strong>.</p>
<p style="text-align: justify;">Pada prinsipnya, ternyata rumus perpangkatan pada bentuk aljabar sama dengan perpangkatan pada bilangan bulat.</p>
<p> </p>
<h2><span style="font-size: 18pt;">Rumus Perpangkatan pada Bentuk Aljabar</span></h2>
<p><img decoding="async" style="width: 600px; display: block; margin: 0px auto;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/rumus%20perpangkatan%20bentuk%20aljabar.png" alt="rumus perpangkatan bentuk aljabar" width="600" /></p>
<p> </p>
<h2><span style="font-size: 18pt;">Bentuk Istimewa dalam Perpangkatan Bentuk Aljabar</span></h2>
<p style="text-align: justify;">Selain itu, terdapat beberapa bentuk istimewa yang akan sering kamu temui dalam perpangkatan aljabar, yaitu:</p>
<p><img decoding="async" style="width: 600px; display: block; margin: 0px auto;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/bentuk%20istimewa%20perpangkatan%20aljabar.png" alt="bentuk istimewa perpangkatan aljabar" width="600" /></p>
<p> </p>
<h2><span style="font-size: 18pt;">Contoh Soal Perpangkatan Bentuk Aljabar</span></h2>
<p style="text-align: justify;">Sekarang, yuk, coba kerjakan soal di bawah ini supaya kamu semakin paham!</p>
<p> </p>
<p><strong>Contoh 1</strong></p>
<p><strong><img decoding="async" style="width: 269px; display: block; margin: 0px auto;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/pa2.png" alt="perpangkatan bentuk aljabar" width="269" /></strong></p>
<p> </p>
<p><strong>Penyelesaian</strong>:<br />
<img decoding="async" style="width: 572px; display: block; margin: 0px auto;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/pa3.png" alt="perpangkatan bentuk aljabar" width="572" /></p>
<p> </p>
<p><strong>Baca Juga: <a href="https://www.ruangguru.com/blog/matematika-kelas-7-mengenal-operasi-hitung-pada-pecahan">Mengenal Operasi Hitung pada Pecahan</a></strong></p>
<p> </p>
<p><strong>Contoh 2</strong></p>
<p><span style="color: #0600ff;"><strong><img decoding="async" style="width: 485px; display: block; margin: 0px auto;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/pa4.png" alt="perpangkatan bentuk aljabar" width="485" /></strong></span></p>
<p> </p>
<p><strong>Penyelesaian:</strong></p>
<p style="text-align: justify;">Nah, untuk contoh yang satu ini, penyelesaiannya cukup rumit nih. Jadi, simak baik-baik ya langkah-langkahnya. Oke, langkah pertama yang bisa kamu lakukan untuk menyelesaikan soal di atas adalah operasikan persamaan tersebut seperti biasa.</p>
<p><img decoding="async" style="width: 388px; display: block; margin: 0px auto;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/pa5.png" alt="perpangkatan bentuk aljabar" width="388" /></p>
<p> </p>
<p>Kemudian, langkah keduanya adalah kita samakan variabelnya.</p>
<p><img decoding="async" style="width: 443px; display: block; margin: 0px auto;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/pa6.png" alt="perpangkatan bentuk aljabar" width="443" /></p>
<p> </p>
<p style="text-align: justify;">Berdasarkan persamaan (3), didapat nilai q = 4. Lalu, kita substitusikan nilai q ke persamaan (2) menjadi sebagai berikut:</p>
<p><img decoding="async" style="width: 140px; display: block; margin: 0px auto;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/pa7.png" alt="perpangkatan bentuk aljabar" width="140" /></p>
<p> </p>
<p style="text-align: justify;">Setelah kita substitusikan niai q ke persamaan (2), didapat nilai p = 5. Selanjutnya, kita substitusikan nilai p ke persamaan (1) menjadi sebagai berikut:</p>
<p><img decoding="async" style="width: 92px; display: block; margin: 0px auto;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/pa8.png" alt="perpangkatan benuk aljabar" width="92" /><br />
Sehingga, diperoleh nilai r adalah 10.</p>
<p style="text-align: justify;">Bagaimana sampai di sini? Paham atau paham? Bagi kamu yang masih kurang paham, jangan ragu untuk tuliskan pertanyaanmu di kolom komentar, ya.</p>
<p style="text-align: justify;">Berdasarkan penjabaran di atas, sekarang kamu sudah tahu bagaimana cara menyelesaikan operasi perpangkatan pada bentuk aljabar berpangkat dua, kan.</p>
<p style="text-align: justify;">Lalu, bagaimana cara untuk menyelesaikan operasi perpangkatan pada bentuk aljabar yang pangkatnya lebih dari dua? Jawabannya ada dua cara, nih. Penasaran bagaimana caranya? Kalau begitu, yuk langsung simak contohnya di bawah ini!</p>
<p><img decoding="async" style="width: 161px; display: block; margin: 0px auto;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/pa9.png" alt="perpangkatan bentuk aljabar" width="161" /></p>
<p> </p>
<p><strong>Penyelesaian:</strong></p>
<ul>
<li><strong>Cara 1: Menggunakan pola segitiga pascal</strong></li>
</ul>
<p>Sebelumnya, ayo kita perhatikan ilustrasi gambar pola segitiga pascal berikut ini dulu, ya!<br />
<img decoding="async" style="width: 260px; display: block; margin: 0px auto;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/segitiga%20paskal%20bentuk%20aljabar.gif" alt="segitiga paskal bentuk aljabar" width="260" /></p>
<p style="text-align: center;"><em>Segitiga pascal (sumber:en.wikipedia.org)</em></p>
<p> </p>
<p style="text-align: justify;">Pada gambar di atas dapat kamu ketahui bahwa, pola bilangan tersebut membentuk bangun segitiga yang selalu diawali dan diakhiri dengan angka 1.</p>
<p style="text-align: justify;">Kemudian, bilangan-bilangan yang selain angka 1 itu diperoleh dari jumlah dua buah bilangan yang terletak di atasnya dan saling berdekatan.</p>
<p style="text-align: justify;"><strong>Pola segitiga pascal ini ternyata berhubungan dengan koefisien pada bentuk aljabar</strong>, <em>lho</em>. Coba kamu perhatikan gambar di bawah ini!</p>
<p><img decoding="async" style="width: 600px; display: block; margin: 0px auto;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/perpangkatan%20bentuk%20aljabar.png" alt="pola segitiga pascal" width="600" /></p>
<p> </p>
<p style="text-align: justify;">Ternyata, <strong>koefisien pada bentuk aljabar</strong> di atas <strong>sama dengan pola segitiga pascal</strong>. Selanjutnya, coba kamu perhatikan besar pangkat pada masing-masing variabelnya.</p>
<p style="text-align: justify;">Ternyata, <strong>semakin ke kanan, besar pangkat variabel a akan semakin kecil</strong> (a<sup>n</sup> –> a<sup>n-1</sup> –> … –> a<sup>0</sup>) dan <strong>besar pangkat variabel b akan semakin besar</strong> (b<sup>0</sup> –> … –> b<sup>n</sup>).</p>
<p style="text-align: justify;">Jadi, berdasarkan soal di atas dapat kita peroleh hasil sebagai berikut:</p>
<p><img decoding="async" style="width: 414px; display: block; margin: 0px auto;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/pa10.png" alt="perpangkatan bentuk aljabar" width="414" /></p>
<p> </p>
<p style="text-align: justify;">Paham ya dengan cara segitiga pascal ini. Oke, kalau begitu kita lanjut ke cara yang kedua, ya.<span style="background-color: transparent;"> </span></p>
<p><strong>Baca Juga: <a href="https://www.ruangguru.com/blog/jenis-jenis-bilangan-pecahan">Jenis-Jenis Bilangan Pecahan dan Contohnya</a></strong></p>
<p> </p>
<ul>
<li><strong>Cara 2:</strong></li>
</ul>
<p style="text-align: justify;">Nah, bagi kamu yang malas menghafal pola segitiga pascal di atas, kamu bisa menggunakan cara yang satu ini. Pada cara ini, kamu hanya tinggal menguraikan aljabar tersebut menjadi sebagai berikut:</p>
<p><img decoding="async" style="width: 492px; display: block; margin: 0px auto;" src="https://cdn-web.ruangguru.com/landing-pages/assets/hs/p12.png" alt="perpangkata bentuk aljabar" width="492" /></p>
<p> </p>
<p style="text-align: justify;">Wah, lumayan panjang ya. Mungkin, untuk menyelesaikan bentuk aljabar dengan pangkat 3 dan 4, cara ini masih cukup efektif. Tapi, bagaimana dengan bentuk aljabar yang pangkatnya 5, 6, 7, dan seterusnya?</p>
<p style="text-align: justify;">Jangan sampai waktu kamu habis hanya untuk menyelesaikan bentuk aljabar yang seperti itu, ya.</p>
<p style="text-align: justify;">Oleh karena itu, ada pentingnya juga <em>nih</em> untuk selalu mengingat pola segitiga pascal yang telah kita pelajari sebelumnya. Selain itu, dalam mengerjakan soal-soal aljabar ini juga dibutuhkan ketelitian yang tinggi.</p>
<p style="text-align: justify;"><strong>Perhatikan selalu tanda (+) dan (-)</strong> nya karena kedua tanda itu akan berubah ketika kamu melakukan operasi perkalian dan pemangkatan.</p>
<p style="text-align: center;">—</p>
<p style="text-align: justify;"><em>So</em>, bagaimana tanggapanmu setelah membaca artikel ini? Mudah atau sulit?</p>
<p style="text-align: justify;">Perlu kita akui kalau materi aljabar ini memang cukup rumit, ya. Oleh karena itu, kamu juga perlu untuk memperbanyak latihan soal, nih.</p>
<p style="text-align: justify;">Nah, bagi kamu yang masih belum paham dengan materi ini dan ingin tanya-tanya lebih lanjut lagi, kamu bisa lho gabung dengan <strong><a href="https://www.ruangguru.com/privat" target="_blank" rel="noopener">Ruangguru Privat</a></strong>. Belajar jadi semakin asik karena kamu bisa berdiskusi dengan kakak-kakak Tutor lulusan PTN terbaik di Indonesia!</p>
<p><a href="https://cta.ruangguru.com/blog-rg-10-privat-b"><img decoding="async" class="aligncenter size-medium" src="https://cdn-web-2.ruangguru.com/landing-pages/assets/84b6f2c7-9c9c-480d-a218-7961334af9ce.png" alt="CTA Ruangguru Privat" /></a></p>
<p style="text-align: left;"><strong>Referensi:</strong></p>
<p style="text-align: left;">Manik D.R, (2009) Penunjang Belajar Matematika untuk SMP/MTs Kelas 7. Jakarta: Pusat Perbukuan, Departemen Pendidikan Nasional.</p>
<p style="text-align: left;"><strong>Sumber Foto:</strong></p>
<p style="text-align: left;">Ilustrasi ‘Segitiga Pascal’ [Daring]. Tautan: https://upload.wikimedia.org/wikipedia/commons/0/0d/PascalTriangleAnimated2.gif (Diakses: 28 Desember 2020)</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>