Kalkulator Punca Kuasa Tiga
Cari ∛x untuk sebarang nombor termasuk nombor negatif
Masukkan sebarang nombor positif atau negatif untuk mencari punca kuasa tiganya.
∛
Punca Kuasa Tiga Kalkulator
Kira punca kuasa tiga bagi sebarang nombor — positif atau negatif — serta-merta dengan langkah pengiraan dan bentuk radikal dipermudahkan. 100% percuma.
segera Langkah demi langkah 100% Percuma
Kira Segera — Punca Kuasa Tiga Klik nombor untuk hasil segera
Instant ∛ Calculation Simplified Radical Form Step-by-Step Breakdown 100% Free & Unlimited
Jadual Kandungan 16 Bahagian Tunjukkan Sembunyikan
Panduan Lengkap
Apakah Itu Punca Kuasa Tiga (Racine Cubique)?
Punca kuasa tiga merupakan operasi utama dalam geometri 3 dimensi. Kalkulator kami memberikan penyelesaian tepat serta-merta.
Punca kuasa tiga ialah nombor yang apabila didarabkan dengan dirinya sebanyak 3 kali menghasilkan nombor asal (3 × 3 × 3 = 27, maka ∛27 = 3).
Radical Anatomy Explorer Click or hover over any part
³ √ 27 = 3
Index (Degree 3)
The small number 3 in the upper-left of the radical sign. It indicates you are finding the 3rd root (cube root) rather than a square root (index 2).
Radical Symbol
The mathematical sign ∛ denoting the extraction of the cube root. In Unicode, it is character U+221B.
Radicand
The number underneath the radical sign whose cube root is being calculated. It can be positive, negative, or zero.
Cube Root (Result)
The real value that, when multiplied by itself three times (y × y × y), equals the radicand.
∛8 = 2 kerana 2 × 2 × 2 = 8
∛64 = 4 kerana 4 × 4 × 4 = 64
∛1000 = 10 kerana 10 × 10 × 10 = 1,000
Ia adalah asas untuk mengira panjang sisi kiub daripada isi padunya.
Arahan
Cara Menggunakan Kalkulator Punca Kuasa Tiga
Tiga langkah mudah:
1
Masukkan Nombor
Taipkan nombor dalam ruangan di atas.
2
Klik Kira
Tekan butang untuk memproses.
3
Semak Jawapan
Ketahui nilai perpuluhan tepat dan bentuk surd, termasuk nombor negatif.
Alat ini mengira semua langkah secara automatik tanpa sebarang rumus rumit.
Notasi
Notasi Matematik: Simbol ∛
Simbol ∛ menandakan punca kuasa tiga dengan indeks darjah 3.
∛64 = 4
Standard mathematical radical notation using index 3 and radical sign.
64^(1/3) = 4
Fractional power representation: raising any number to 1/3 equals its cube root.
Math.cbrt(64) // 4
Standard in JavaScript, Python (64 ** (1/3)), C, Java, and modern runtimes.
=POWER(A1, 1/3)
Excel and Google Sheets formula. Alternatively use =A1^(1/3).
\sqrt[3]64
LaTeX mathematical typesetting command for academic papers and documents.
Unicode: U+221B | HTML: ∛ | 221B + Alt+X
On Windows: type 221B then press Alt+X. On Mac: Cmd+Ctrl+Space. HTML: ∛.
Ia juga boleh ditulis dalam bentuk indeks pecahan:
x^(1/3) = ∛x
Kedua-duanya sama. Dalam pengaturcaraan diguna x^(1/3).
Kedua-dua ∛x dan x^(1/3) merujuk kepada operasi yang sama.
Formula
Definisi Punca Kuasa Tiga
Punca kuasa tiga ditakrifkan oleh songsangan kuasa tiga:
If y = ∛x, then y × y × y = x (or y³ = x)
Didarabkan tiga kali dengan dirinya, hasil mesti mengembalikan nombor asal.
Hukum Asas
Lima Hukum Punca Kuasa Tiga
Hukum memudahkan pengendalian radikal kuasa 3:
1
Hukum Hasil Darab
Memisahkan faktor radikan
∛8 × ∛27 = 2 × 3 = 6 = ∛216
Memudahkan pemfaktoran nombor besar.
2
Hukum Hasil Bahagi
Memisahkan pengangka dan penyebut
∛64 ÷ ∛8 = 4 ÷ 2 = 2 = ∛8
Membolehkan penarikan punca pecahan berasingan.
3
Hukum Kuasa
Kuasa tiga dan punca membatalkan satu sama lain
∛(5³) = 5 dan ∛((-3)³) = -3
Sah untuk semua nombor nyata, positif mahupun negatif!
4
Penaikan Kuasa Tiga
Kuasa 3 bagi punca memulangkan radikan
(∛5)³ = 5
Penting dalam menyelesaikan persamaan radikal.
5
Nilai Tetap
Punca bagi 0 dan 1 tidak berubah
∛1 = 1 dan ∛0 = 0
Titik rujukan analisis matematik.
Working with higher degree root indices beyond 3? Compute 4th, 5th, and general nth roots online →
Kaedah
Kaedah Mengira Punca Kuasa Tiga
Empat kaedah umum:
Method 1
1. Kalkulator Atas Talian (Paling Pantas)
Masukkan nombor dan peroleh jawapan serta-merta.
Method 2
2. Pemfaktoran Perdana
Kumpulkan faktor perdana dalam set 3 nombor.
Contoh: Mengira ∛216
- 216 = 2³ × 3³
- Set tiga: (2×2×2) × (3×3×3)
- Punca: 2 × 3 = 6
Extended Example: Find ∛3375
- Decompose into prime factors: 3375 = 3 × 3 × 3 × 5 × 5 × 5
- Group into sets of three: (3 × 3 × 3) × (5 × 5 × 5) = 3³ × 5³
- Take one factor from each group: 3 × 5 = 15
- Therefore, ∛3375 = 15.
Method 3
4. Kaedah Anggaran
Bandingkan di antara dua kuasa tiga terdekat.
Contoh: Anggaran ∛50
- 3³ = 27
- 4³ = 64
- ∛50 berada di antara 3 dan 4 (kira-kira 3.68).
Kuasa Tiga Sempurna
Senarai Kuasa Tiga Sempurna Penting
Kuasa tiga membesar sangat laju. Berikut 10 yang pertama:
1³ = 1
∛1 = 1
2³ = 8
∛8 = 2
3³ = 27
∛27 = 3
4³ = 64
∛64 = 4
5³ = 125
∛125 = 5
6³ = 216
∛216 = 6
7³ = 343
∛343 = 7
8³ = 512
∛512 = 8
9³ = 729
∛729 = 9
10³ = 1,000
∛1,000 = 10
11³ = 1,331
∛1,331 = 11
12³ = 1,728
∛1,728 = 12
Tatal untuk melihat jadual ↓
| Number (n) | Cube (n³) | Cube Root (∛n³) | Quick Test |
|---|---|---|---|
| 1 | 1 | 1 | |
| 2 | 8 | 2 | |
| 3 | 27 | 3 | |
| 4 | 64 | 4 | |
| 5 | 125 | 5 | |
| 6 | 216 | 6 | |
| 7 | 343 | 7 | |
| 8 | 512 | 8 | |
| 9 | 729 | 9 | |
| 10 | 1,000 | 10 | |
| 11 | 1,331 | 11 | |
| 12 | 1,728 | 12 | |
| 15 | 3,375 | 15 | |
| 20 | 8,000 | 20 | |
| 25 | 15,625 | 25 | |
| 50 | 125,000 | 50 | |
| 100 | 1,000,000 | 100 |
Nombor Negatif
Punca Nombor Negatif (Ianya Nyata!)
Perbezaan paling utama daripada punca kuasa dua: punca kuasa tiga bagi nombor negatif ialah nombor nyata biasa.
Pendaraban tiga nombor negatif menghasilkan nilai negatif: negatif × negatif = positif, positif × negatif = negatif.
Square Root (Index 2)
√(-16) = No Real Root
Requires the imaginary unit i (±4i) because squaring any real number produces a non-negative result.
Cube Root (Index 3)
∛(-64) = -4
100% Real solution! Multiplying three negative numbers gives a negative product: (-4) × (-4) × (-4) = -64.
Contoh: ∛(−64)
∛(−64) = −4, kerana (−4) × (−4) × (−4) = −64
∛(-x) = -∛x
Tidak memerlukan unit khayalan i: jawapannya 100% nyata.
Perbezaan dengan Punca Kuasa Dua
Punca kuasa dua negatif memerlukan i, manakala punca kuasa tiga menghasilkan nombor nyata terus.
Cube Root of Decimals & Cube Root of Fractions
Cube Root of Decimals & Cube Root of Fractions
Cube Root of Decimals
To find the cube root of a decimal, convert it into a fraction or power of ten, simplify, and evaluate the root.
Example: ∛0.008
- 0.008 = 8 / 1000 = 8 × 10⁻³
- ∛(8 / 1000) = ∛8 / ∛1000 = 2 / 10
- 2 / 10 = 0.2
Cube Root of Fractions
Apply the quotient property by taking the cube root of the numerator and denominator separately: ∛(a/b) = ∛a / ∛b.
Example: ∛(8 / 27)
- Apply quotient rule: ∛(8 / 27) = ∛8 / ∛27
- Evaluate numerator: ∛8 = 2
- Evaluate denominator: ∛27 = 3
- Result: 2 / 3 ≈ 0.6667
Pemudahan
Jadual Pemudahan Punca Kuasa Tiga
Keluarkan kuasa tiga sempurna dari bawah punca:
| Ungkapan | Bentuk Termudah | Sebab | Actions |
|---|---|---|---|
| ∛16 | 2∛2 | 16 = 2³ × 2 = 8 × 2 | |
| ∛24 | 2∛3 | 24 = 2³ × 3 = 8 × 3 | |
| ∛32 | 2∛4 | 32 = 2³ × 4 = 8 × 4 | |
| ∛40 | 2∛5 | 40 = 2³ × 5 = 8 × 5 | |
| ∛48 | 2∛6 | 48 = 2³ × 6 = 8 × 6 | |
| ∛54 | 3∛2 | 54 = 3³ × 2 = 27 × 2 | |
| ∛56 | 2∛7 | 56 = 2³ × 7 = 8 × 7 | |
| ∛72 | 2∛9 | 72 = 2³ × 9 = 8 × 9 | |
| ∛108 | 3∛4 | 108 = 3³ × 4 = 27 × 4 | |
| ∛128 | 4∛2 | 128 = 4³ × 2 = 64 × 2 | |
Cari nombor kuasa tiga sempurna terbesar yang membahagi radikan.
Perbandingan
Punca Kuasa Tiga vs Punca Kuasa Dua
Perbandingan 3D lawan 2D:
| Property / Feature | Punca Kuasa Tiga | Punca Kuasa Dua |
|---|---|---|
| Simbol | ∛ (Index 3) | √ (Index 2) |
| Exponent Form | x^(1/3) | x^(1/2) |
| Nombor negatif | Ada jawapan nyata (∛-64 = -4) | Tiada jawapan nyata (perlu i) |
| Bilangan pendaraban | 3 (x × x × x) | 2 (x × x) |
| Number of Real Roots | Exactly 1 for all real numbers | 1 for positive / 0 for negative |
| Graph Symmetry | Odd function (symmetric about origin) | Defined only for x ≥ 0 |
| Output Sign | Always matches input sign (∛(-8) = -2) | Always non-negative (principal root) |
| Contoh | ∛64 = 4 (4³=64) | √64 = 8 (8²=64) |
Need to evaluate square roots instead? Calculate square roots with exact decimals and simplified radicals →
Jadual Rujukan
Jadual Lengkap Punca Kuasa Tiga 1 hingga 100
Senarai nilai tepat kepada 3 tempat perpuluhan.
Tatal untuk melihat semua 100 nilai ↓
| Nombor (n) | Punca (∛n) | Simplified Radical | Jenis | Quick Test |
|---|---|---|---|---|
| 1 | 1 | 1 | Kuasa Tiga Sempurna | |
| 2 | 1.2599 | ∛2 | Bukan Nisbah | |
| 3 | 1.4422 | ∛3 | Bukan Nisbah | |
| 4 | 1.5874 | ∛4 | Bukan Nisbah | |
| 5 | 1.7100 | ∛5 | Bukan Nisbah | |
| 6 | 1.8171 | ∛6 | Bukan Nisbah | |
| 7 | 1.9129 | ∛7 | Bukan Nisbah | |
| 8 | 2 | 2 | Kuasa Tiga Sempurna | |
| 9 | 2.0801 | ∛9 | Bukan Nisbah | |
| 10 | 2.1544 | ∛10 | Bukan Nisbah | |
| 11 | 2.2240 | ∛11 | Bukan Nisbah | |
| 12 | 2.2894 | ∛12 | Bukan Nisbah | |
| 13 | 2.3513 | ∛13 | Bukan Nisbah | |
| 14 | 2.4101 | ∛14 | Bukan Nisbah | |
| 15 | 2.4662 | ∛15 | Bukan Nisbah | |
| 16 | 2.5198 | 2∛2 | Bukan Nisbah | |
| 17 | 2.5713 | ∛17 | Bukan Nisbah | |
| 18 | 2.6207 | ∛18 | Bukan Nisbah | |
| 19 | 2.6684 | ∛19 | Bukan Nisbah | |
| 20 | 2.7144 | ∛20 | Bukan Nisbah | |
| 21 | 2.7589 | ∛21 | Bukan Nisbah | |
| 22 | 2.8020 | ∛22 | Bukan Nisbah | |
| 23 | 2.8439 | ∛23 | Bukan Nisbah | |
| 24 | 2.8845 | 2∛3 | Bukan Nisbah | |
| 25 | 2.9240 | ∛25 | Bukan Nisbah | |
| 26 | 2.9625 | ∛26 | Bukan Nisbah | |
| 27 | 3 | 3 | Kuasa Tiga Sempurna | |
| 28 | 3.0366 | ∛28 | Bukan Nisbah | |
| 29 | 3.0723 | ∛29 | Bukan Nisbah | |
| 30 | 3.1072 | ∛30 | Bukan Nisbah | |
| 31 | 3.1414 | ∛31 | Bukan Nisbah | |
| 32 | 3.1748 | 2∛4 | Bukan Nisbah | |
| 33 | 3.2075 | ∛33 | Bukan Nisbah | |
| 34 | 3.2396 | ∛34 | Bukan Nisbah | |
| 35 | 3.2711 | ∛35 | Bukan Nisbah | |
| 36 | 3.3019 | ∛36 | Bukan Nisbah | |
| 37 | 3.3322 | ∛37 | Bukan Nisbah | |
| 38 | 3.3620 | ∛38 | Bukan Nisbah | |
| 39 | 3.3912 | ∛39 | Bukan Nisbah | |
| 40 | 3.4200 | 2∛5 | Bukan Nisbah | |
| 41 | 3.4482 | ∛41 | Bukan Nisbah | |
| 42 | 3.4760 | ∛42 | Bukan Nisbah | |
| 43 | 3.5034 | ∛43 | Bukan Nisbah | |
| 44 | 3.5303 | ∛44 | Bukan Nisbah | |
| 45 | 3.5569 | ∛45 | Bukan Nisbah | |
| 46 | 3.5830 | ∛46 | Bukan Nisbah | |
| 47 | 3.6088 | ∛47 | Bukan Nisbah | |
| 48 | 3.6342 | 2∛6 | Bukan Nisbah | |
| 49 | 3.6593 | ∛49 | Bukan Nisbah | |
| 50 | 3.6840 | ∛50 | Bukan Nisbah | |
| 51 | 3.7084 | ∛51 | Bukan Nisbah | |
| 52 | 3.7325 | ∛52 | Bukan Nisbah | |
| 53 | 3.7563 | ∛53 | Bukan Nisbah | |
| 54 | 3.7798 | 3∛2 | Bukan Nisbah | |
| 55 | 3.8030 | ∛55 | Bukan Nisbah | |
| 56 | 3.8259 | 2∛7 | Bukan Nisbah | |
| 57 | 3.8485 | ∛57 | Bukan Nisbah | |
| 58 | 3.8709 | ∛58 | Bukan Nisbah | |
| 59 | 3.8930 | ∛59 | Bukan Nisbah | |
| 60 | 3.9149 | ∛60 | Bukan Nisbah | |
| 61 | 3.9365 | ∛61 | Bukan Nisbah | |
| 62 | 3.9579 | ∛62 | Bukan Nisbah | |
| 63 | 3.9791 | ∛63 | Bukan Nisbah | |
| 64 | 4 | 4 | Kuasa Tiga Sempurna | |
| 65 | 4.0207 | ∛65 | Bukan Nisbah | |
| 66 | 4.0412 | ∛66 | Bukan Nisbah | |
| 67 | 4.0615 | ∛67 | Bukan Nisbah | |
| 68 | 4.0817 | ∛68 | Bukan Nisbah | |
| 69 | 4.1016 | ∛69 | Bukan Nisbah | |
| 70 | 4.1213 | ∛70 | Bukan Nisbah | |
| 71 | 4.1408 | ∛71 | Bukan Nisbah | |
| 72 | 4.1602 | 2∛9 | Bukan Nisbah | |
| 73 | 4.1793 | ∛73 | Bukan Nisbah | |
| 74 | 4.1983 | ∛74 | Bukan Nisbah | |
| 75 | 4.2172 | ∛75 | Bukan Nisbah | |
| 76 | 4.2358 | ∛76 | Bukan Nisbah | |
| 77 | 4.2543 | ∛77 | Bukan Nisbah | |
| 78 | 4.2727 | ∛78 | Bukan Nisbah | |
| 79 | 4.2908 | ∛79 | Bukan Nisbah | |
| 80 | 4.3089 | 2∛10 | Bukan Nisbah | |
| 81 | 4.3267 | 3∛3 | Bukan Nisbah | |
| 82 | 4.3445 | ∛82 | Bukan Nisbah | |
| 83 | 4.3621 | ∛83 | Bukan Nisbah | |
| 84 | 4.3795 | ∛84 | Bukan Nisbah | |
| 85 | 4.3968 | ∛85 | Bukan Nisbah | |
| 86 | 4.4140 | ∛86 | Bukan Nisbah | |
| 87 | 4.4310 | ∛87 | Bukan Nisbah | |
| 88 | 4.4480 | 2∛11 | Bukan Nisbah | |
| 89 | 4.4647 | ∛89 | Bukan Nisbah | |
| 90 | 4.4814 | ∛90 | Bukan Nisbah | |
| 91 | 4.4979 | ∛91 | Bukan Nisbah | |
| 92 | 4.5144 | ∛92 | Bukan Nisbah | |
| 93 | 4.5307 | ∛93 | Bukan Nisbah | |
| 94 | 4.5468 | ∛94 | Bukan Nisbah | |
| 95 | 4.5629 | ∛95 | Bukan Nisbah | |
| 96 | 4.5789 | 2∛12 | Bukan Nisbah | |
| 97 | 4.5947 | ∛97 | Bukan Nisbah | |
| 98 | 4.6104 | ∛98 | Bukan Nisbah | |
| 99 | 4.6261 | ∛99 | Bukan Nisbah | |
| 100 | 4.6416 | ∛100 | Bukan Nisbah |
Want to verify whether any specific integer is an exact cube? Test integers with our instant perfect cube checker →
Aplikasi
Aplikasi Nyata
Punca kuasa tiga sangat penting dalam konteks 3 dimensi:
Seni Bina & 3D
Mencari sisi bilik kiub: bilik 64 m³ mempunyai sisi ∛64 = 4 m.
Logistik & Pakej
Dimensi kotak dan berat volumetrik parcel.
Fizik & Kejuruteraan
Jejari sfera daripada isi padu dan termodinamik.
Kewangan & CAGR
Kadar pertumbuhan tahunan 3 tahun: r = ∛(Akhir / Awal) - 1.
Ukuran Harian
Panjang sisi tangki kiub 27 liter (30 cm).
Under the Hood
How Does This Cube Root Calculator Work?
1
Client-Side High Performance
All calculations execute directly in your browser with zero latency and 100% data privacy.
2
Exact Integer Detection
Instantly checks whether the radicand is a perfect cube and outputs exact integers without rounding drift.
3
Radical Simplification Engine
Performs prime factorization in real time to extract triple factors, rendering exact forms like ∛24 = 2∛3.
4
Full Real Negative Support
Applies the mathematical odd symmetry rule ∛(-x) = -∛x, returning exact real solutions for negative inputs.
5
Arbitrary Precision & Step Workings
Provides clean decimal approximations and comprehensive step-by-step mathematical reasoning.
Soalan Lazim
Soalan Lazim
What is a cube root?
A cube root of a number is a value that, when multiplied by itself three times, produces the original number. In mathematical notation, the cube root of x is written as ∛x or x^(1/3). For example, the cube root of 27 is 3, because 3 × 3 × 3 = 27.
Can you take the cube root of a negative number?
Yes! Unlike square roots, cube roots of negative numbers are real numbers because multiplying three negative numbers results in a negative product. The rule is ∛(-x) = -∛x. For example, ∛(-8) = -2, because (-2) × (-2) × (-2) = -8.
What is the difference between cubing and taking a cube root?
Cubing and taking the cube root are inverse operations. Cubing a number means raising it to the power of 3 (for example, 4³ = 64). Taking the cube root reverses this process to recover the original base value: ∛64 = 4.
What is a perfect cube?
A perfect cube is an integer that results from multiplying an integer by itself three times. Examples include 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000. The cube root of a perfect cube is always a whole number.
How do you calculate a cube root without a calculator?
For perfect cubes, use prime factorization: break the number into prime factors, group identical factors into sets of three, and extract one from each group (e.g. 3375 = 3³ × 5³ = 15). For non-perfect cubes, use estimation between the two nearest known perfect cubes.
What is the cube root symbol?
The cube root symbol is ∛ (Unicode character U+221B). It consists of a standard radical sign with a small index 3 in the upper-left notch. In software code it is written cbrt(x), and in LaTeX as \sqrt[3]{x}.
How do you type the cube root symbol?
On Windows, type 221B then press Alt+X in supported apps, or use Character Map. On Mac, press Cmd+Ctrl+Space to open Character Viewer and search 'cube root'. In HTML, use the entity ∛ or copy and paste ∛ directly.
What is ∛0?
The cube root of 0 is exactly 0, because 0 × 0 × 0 = 0.
Is the cube root of 2 rational or irrational?
The cube root of 2 (∛2 ≈ 1.2599) is an irrational number. It cannot be expressed as an exact fraction of two integers, and its decimal expansion continues infinitely without repeating.
What is the cube root of 1?
The real cube root of 1 is 1, since 1 × 1 × 1 = 1. (In complex mathematics, there are also two non-real complex cube roots of unity: e^(2πi/3) and e^(-2πi/3)).
Rumusan
Kesimpulan Ringkas
Fakta
Punca kuasa tiga menjawab: nombor apakah yang didarabkan 3 kali menghasilkan nilai ini? Selesaikan segera dengan kalkulator kami di atas.