Cube Root

The cube root of a number is that number which is multiplied 3 times to get the original number. Whenever a number (x) is multiplied three times, then the resultant number is known as the cube of that number. The cube for the number (x) is represented as x³ and is read as 'x-cubed' (or) 'x to the power of 3' (or) 'x raised to 3. For example, let us take the number 5. We know that 5 × 5 × 5 = 125. Hence, 125 is called the cube of 5. While on the other hand, finding the cube root of a number involves the reverse process of the cube of a number and is denoted by ∛. Considering the same example, 5 is called the cube root of the number 125 and is written as ∛125 = 5. On this page, we will learn more about the cubes and cube roots of a number.

When we think about the words cube and root, the first picture that might come to our mind is a literal cube and the roots of a tree. Isn't it? Well, the idea is similar. Root means the primary source or origin. So, we just need to think 'cube of which number should be taken to get the given number'. In mathematics, the definition of cube root says that Cube root is the number that needs to be multiplied three times to get the original number. Now, let us look at the cube root formula, where y is the cube root of x. ∛x = y. The radical sign ∛ is used as a cube root symbol for any number with a small 3 written on the top left of the sign. Another way to denote cube root is to write 1/3 as the exponent of a number.

Cube root is an inverse operation of the cube of a number.

Cube Root Symbol

The cube root symbol which is used to represent the cube root of a number is ∛. This means, if we want to write the cube root of 27, we use the cube root symbol in the following way. ∛27 = 3

Perfect Cubes

A perfect cube is an integer that can be expressed as the product of three equal integers. For example, 125 is a perfect cube because 5³ = 5 × 5 × 5 = 125. However, 121 is not a perfect cube because there is no number, which when multiplied three times gives the product 121. In other words, a perfect cube is a number whose cube root is an integer. It is also known as the 'cube number'. The following table shows the perfect cubes of the first 10 natural numbers.

The cube root of a number can be determined by using the prime factorization method. In order to find the cube root of a number:

• Step 1: Start with the prime factorization of the given number.
• Step 2: Then, divide the factors obtained into groups containing three same factors.
• Step 3: After that, remove the cube root symbol and multiply the factors to get the answer. If there is any factor left that cannot be divided equally into groups of three, that means the given number is not a perfect cube and we cannot find the cube root of that number.

Example: Let us see how to find the cube root of 15625.

The cube root formula helps in the calculation of the cube root of any given number that is expressed in a radical form using the symbol ∛. It can be calculated by first finding out the prime factorization of the given number and then later applying the cube root formula. Suppose, x is any number such that, x = y × y × y.

The formula to calculate the cube root is given as:

• Cube root of x = ∛x = ∛(y × y × y) = y

where, y is the cube root of any number x. This also means that the number x would be a perfect cube if y has an integer value.

Applications of Cube Root Formula

Given below are a few major applications of the cube root formula:

• Solve cubic equations.
• Find the dimensions of a cube if the volume is given.

When we multiply a number three times by itself, the resultant number (product) is known as the cube of the original number. We call it a cube because it is used to represent the volume of a cube. In other words, a number raised to exponent 3 is called the cube of that number. For example, the cube of 3 is 27. That means 3 × 3 × 3 = 27, and it can be written as 3³. Similarly, the cube of 4 is 64, and the cube of 5 is 125, and so on.

To find the cube of a number, first, multiply that number by itself, then multiply the product obtained with the original number again. Let us find the cube of 7 through the same process. We know that the cube of a number N is N × N × N. So, the cube of 7 is 7 × 7 × 7. Now, in order to find the cube of 7, we will first find the value of 7 × 7. This value is 49. Now, we will find 49 × 7. This is equal to 343. Hence, we can say that cube of the number 7 is 343.

Cube of a Fraction

Similar to the cube of a number, the cube of a fraction can be found by multiplying it three times. For example, the cube of the fraction (2/5) can be written as 2/5 × 2/5 × 2/5. Simplifying it further, we get the value of the cube as (2 × 2 × 2) / (5 × 5 × 5). This is equal to (2³/ 5³) = 8/125.

Cube of Negative Numbers

The process to find the cube of a negative number is the same as that of a whole number and fraction. Here, always remember that the cube of a negative number is always negative, while the cube of a positive number is always positive. For example, let us try finding the cube of -7. We know that the cube of -7 is (-7) × (-7) × (-7). Now, in order to find the cube of (-7), we will first find the value of (-7) × (-7). This value is 49. Now, we will find 49 × (-7). This is equal to -343. Hence, we can say that cube of the number -7 is -343. The cube root formula for negative numbers is:

• Perform the prime factorization method of the given number.
• Divide the factors obtained into three groups, each of which should contain the same set of factors.
• Multiply the factors in any one group to get the cube root.
• It's just that the product of three negative values gives us a negative result. which is represented by the negative sign with the cube root of a negative number.

☛ Related Topics

Check these interesting articles related to the cube root.

• Cube Root Calculator
• Squares and Square Roots
• Radical

What is Cube Root of a Number?

The cube root is the reverse of the cube of a number and is denoted by ∛. For example, ∛216, that is, the cube root of 216 = 6 because when 6 is multiplied thrice with itself, it gives 216. In other words, since 6³ = 216, we have ∛216 = 6.

What is the Difference Between Cube and Cube Root?

• When a number is multiplied by itself 3 times, then the product is the cube of the given number. For example, cube of 3 is 27 because 3 × 3 × 3 = 27.
• The cube root of a number is a number whose cube is the given number. For example, the cube root of 27 is 3

What is Cube Root Used For?

Cube root is used to solve cubic equations. They are also used to find the side length of a cube if its volume is given.

How to Find the Cube Root of a Number?

Cube root can be simplified using the prime factorization method. First, do the prime factorization of the given number, then take out the common factors in groups of 3. Multiply these common factors taking only one from each group to get the answer.

Is it Possible to Simplify the Negative Cube Root?

Yes, the simplification of negative cube roots is the same as positive cube roots. The only difference is the presence of a negative sign with the cube root of a negative number.

What is Not a Perfect Cube?

A number is not a perfect cube if we cannot make 3 equal groups of factors of the number after doing the prime factorization. For example, 144 is not a perfect cube because there is no number that, when multiplied 3 times with itself, gives 144 as the product. In other words, if the cube root of a number is not an integer, then it is not a perfect cube.

What is the Cube of an Odd Natural Number?

The cube of an odd natural number is always an odd number. For example, 5³ = 125, 7³ = 343, 9³ = 729, etc.

Can the Cube Root of Any Odd Number be Even?

No, the cube root of an odd number is always odd. For example, the cube root of 27 = (27)^1/3 = 3. Here, both 3 and 27 are odd numbers.

How to Calculate Cube Root of Any Number?

The easiest and basic method to find the cube root of any number is the prime factorization method.

What is the Cube Root Formula in Algebra?

In math, the cube root formula is used to represent any number in the form of its cube root, such as for any number x, its cube root will be ³√x = x^1/3. For example, the cube root of 125 is 5 125^1/3 = (5³)^1/3 = 5.

What is the Cube Root Formula for Negative Numbers?

Yes, it is possible to find the cube root of negative numbers. For example, -64 = (-4) × (-4) × (-4). We can write -64 as the product of 3 negative 4's. Thus, ∛-64 = -4 because the product of three negative values gives us a negative result.

How to Use Cube Root Formula?

In order to use the cube root formula, we use the following steps:

• Step 1: Determine the prime factors of the number, say x using the prime factorization method.
• Step 3: Make groups of factors so obtained, where each group has the product of 3 same factors.
• Step 3: Write that in the form of ³√x = ³√(y×y×y), where y corresponds to the cube root of x.

How to Write Cube Root Formula in Words?

The cube root formula can be written in words in the following way. The cube root of any number is the number raised to the power of 1/3.

What is the Cube Root Symbol?

The cube root symbol which is used to represent the cube root of a number is ∛. This means, if we want to write the cube root of 125, we use the cube root symbol in the following way. ∛125 = 5