Cube Root

In Mathematics, when we think of cube root, it means that it is the primary source or origin. A cube root is the number that, when multiplied by itself three times, gives the original number.

Let us see the denotation of the formula of cube root of a number b. ∛b = a. Here, the cube root sign is shown as ³ , so the number written on the top right shows the root of the given number.

For calculating cube roots without using a calculator, refer to the table given below: 

History of Cube Root

The history of cube root can be dated back to ancient civilizations like the Babylonian civilization. In the early 16th century, Italian mathematician Scipione Del Ferro discovered a formula for solving the cubic equation, which was later developed by Niccolò Tartaglia and Gerolamo Cardano.

The calculations and solving of cube roots have evolved over time, with different mathematicians developing various methods to solve cube roots to get accurate answers. For example, Halley’s method is used to find the cube roots of irrational numbers.
 

There are various properties that are applied for cube roots, some of them are listed below:

• Cube Root of a Product: The cube root of a product is equal to the product of the cube roots of the individual numbers. ∛(a × b) = ∛a × ∛b. For example, ∛(8 × 27) = ∛8 × ∛27 = 2 × 3 = 6. 

• Cube Root of a Quotient: The cube root of a quotient equals the quotient of the cube roots of the numerator and denominator. ∛(a / b) = ∛a / ∛b, (where b ≠ 0). For example, ∛(8 / 27) = ∛8 / ∛27 = 2 / 3. 

• Cube Root of a Cube Number: The cube root of a perfect cube gives a whole number. ∛(a³) = a. For example, ∛(5³) = ∛125 = 5. 

• Cube Root of a Negative Number: Unlike square roots, cube roots of negative numbers are real. This is because multiplying a negative number three times still gives a negative result. ∛(–a) = –∛a. For example, ∛(–8) = –2 (because –2 × –2 × –2 = –8). 
   

• Cube Root of Zero: The cube root of zero is always zero.
  
   
• Multiplicative Property of Cube Roots: The cube root of a number multiplied by the cube root of another number equals the cube root of their product. ∛a × ∛b = ∛(a × b). For example, ∛2 × ∛4 = ∛(8) = 2

• Division Property of Cube Roots: The cube root of one number divided by the cube root of another equals the cube root of their quotient. ∛a ÷ ∛b = ∛(a ÷ b). For example, ∛16 ÷ ∛2 = ∛(8) = 2.

There are various types of cube roots and are classified into four main types. There are other classifications as well, but we will discuss these four types. They are:

• Positive Cube Roots
• Negative Cube Roots
• Perfect Cube Roots
• Imperfect Cube Roots

Positive Cube Roots

Positive cube roots are when a cube root of a number is positive, the result is always positive. For example, ∛8 = 2. The example shows that when a cube root of a number is positive, the result of said cube root is always positive.
 

Negative Cube Roots

Negative cube roots are similar to positive cube roots except when the cube root of a given number is negative the result is always negative. For example, ∛-8 = -2. The explanation is that when you multiply -2 three times, it results in -8. Hence, proving the statement that the result of a cube root of a negative number is always negative.

Perfect Cube Roots

Perfect cube roots are when the result of the cube roots do not have any decimal point. For example, the cube root of 1 is 1, the cube root of 8 is 2, the cube root of 27 is 3 and so on. This shows us that all the results of the cube roots are whole numbers and do not have any decimal points, hence being perfect cubes.

Imperfect Cube Roots

Imperfect cube roots are numbers that are not perfect cubes, which means that the results have decimal points or are irrational numbers. For example, the cube root of 46 is approximately 3.583, or the cube root of 300 is approximately 6.7. If we cannot find the perfect cube root of a given number, that means it is an imperfect cube root.
 

• Understanding cube roots: Learning what cube roots are helps students to understand how numbers relate in three dimensions. Cube roots form the basis for deeper mathematical concepts. 
  
   
• Connection Between Square Roots and Cube Roots: By comparing square roots and cube roots, students develop a better understanding of powers and exponents. This comparison strengthens number sense and prepares them for advanced algebra and geometry.
  
   
• Mastering Perfect Cube Roots: Knowing perfect cube roots (like ∛8 = 2 or ∛27 = 3) and using a cube roots list helps students quickly identify and memorize common values, which is useful for solving math problems efficiently.
  
   
• Learning How to Solve and Simplify Cube Roots: Students gain valuable skills in how to solve cube roots, how to simplify cube roots, and how to factor cube roots. These processes make it easier to handle large numbers and algebraic expressions.
  
   
• Using Tools Like Cube Root Calculators: With practice and tools such as a cube roots calculator, students can verify answers, explore more examples, and strengthen accuracy while solving problems involving factoring cube roots or complex cube roots.
  
   
• Applications in Real-World and Advanced Math: Cube roots are used to solve real-life problems in physics, engineering, and data science. Understanding complex cube roots (like cube roots of i) also builds a foundation for higher-level topics such as complex numbers and polynomial equations.

Here are some tips and tricks students can follow on how to master and solve cube roots:

• Understand the Definition: Before solving, always remember the meaning of cube root. The Cube Root of a number is when a number is multiplied by itself three times, resulting in the original number. \(a^3 = b\).
  
   
• Learning Perfect Cubes: It will help the students if they memorize the first few perfect cubes like 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125 and so on. This will help them to solve cube roots of common numbers.
  
   
• Use Prime Factorization: For larger numbers, if the students find it difficult to find the cube roots, they can use prime factorization. You can break down the number into its prime factors and then group them in sets of three.For example:

∛216 = 6

216 = 2³ × 3³

∛216 = 2  × 3 = 6

 

• Utilize Estimation: Sometimes when students want to find a cube root of an imperfect cube, it is best to estimate the nearest perfect cubes to get the result for the imperfect cube root. For example, 33 = 27 and 43 = 64, so 350 is in between the cube roots of 27 and 64.
  
   
• Practice with Rational and Irrational Numbers: Get comfortable solving cube roots of both rational and irrational numbers, as it will help the students to understand how both kinds of numbers work.
  
   
• Use Calculator Wisely: Using normal calculators will not help students as much as they use the scientific calculators. Scientific calculators will help students to get the most accurate values than the normal calculators.
  
   
• Practice Daily: Nothing is perfected without practice. The students should practice solving cube root problems to understand the best method they can use to solve problems efficiently without mistakes. As they say, practice makes perfect.
  
   
• Relate Cube Roots to Real Objects: Display a list of cube roots (like ∛1 = 1, ∛8 = 2, ∛27 = 3, etc.) in the classroom or at home. Repetition and visual reminders help students quickly recall perfect cube roots.
  
   
• Connect Cube Roots with Square Roots: Show how square roots and cube roots are similar in concept but different in degree. Comparing them builds a strong foundation for understanding exponents and radicals.
  
   
• Teach How to Simplify and Factor Cube Roots: Guide students through examples of how to simplify cube roots and how to factor cube roots. Breaking numbers into prime factors helps them grasp how cube roots are derived.
  
   
• Introduce Cube Roots through Patterns: Ask students to find patterns in cubes (1³ = 1, 2³ = 8, 3³ = 27…). Recognizing these patterns improves their ability to identify perfect cubes and estimate cube roots quickly.

Cube roots have various applications apart from just mathematics. In this article, we will learn more about the real world applications of cube roots.

• Architecture & Construction: Engineers use cube roots to calculate dimensions of water tanks, storage containers, or building blocks.

• Material Science: Cube roots are applied to calculate density-related measurements (mass, volume, and side lengths).

• Shipping & Packaging: To design cubic cartons and containers, cube roots help determine the side when only volume capacity is given.

• Chemistry: Used in molecular structure problems (e.g., cube root of atomic/molecular volumes in crystallography).

• Physics: If volume is cubic, cube root helps in reversing volume to get side length in experiments.