Cube and Cube Root

Whenever a number (x) is multiplied three times, then the resultant answer is known as the cube of that number. Thus, the cube for the number (x) becomes x³ or x-cubed. For example, let us take the number 3. We know that 5 × 5 × 5 = 125. Hence, 125 is called the cube of 5.

While on the other hand, the cube root of a number is the reverse process of the cube of a number and is denoted by ∛ . Considering the same example, 3 is called the cube root of the number 27. In this page, we will learn more about the cube and cube root of a number.

When we multiply a number two times by itself, we get the product of 3 numbers, all three numbers being same. 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 cube as (2 × 2 × 2) / (5 × 5 × 5). This will be 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. 

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 is written as, "Cube root is the number that needs to be multiplied three times to itself to get the original number." Now, let us look at the cube root formula, where y is the cube root of x. \(\sqrt[3]{x}=y\). The radical sign ∛ is used as a cube root symbol for any number with a small 3 written on the left of the sign. Another way to denote cube root is to write 1/3 as the exponent of a number.

It is an inverse operation of the cube of a number.

Perfect Cubes:

A perfect cube is an integer that can be expressed as the product of three same or 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 by itself gives the product 121. 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, start by the prime factorization of the given number. Then, divide the factors obtained into three groups containing the same number of each factor. After that, multiply the factors to get the answer. If there is any factor left that cannot be divided equally into three groups, that means the given number is not a perfect cube. For example, let us see how we find the cube root of 15625.

What are Cubes and Cube Roots?

The cube of a number is the value of the third exponent of the number. For example, the cube of 2 is 2³= 2 × 2 × 2 = 8. While the cube root is the reverse process 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.

What are Cube Roots Used For?

Cube roots are used to solve cubic equations. They are also used to find the dimensions of a cube if the volume is given.

How Do You Simplify Cube Roots?

Cube roots 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 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 of the number. For example, 121 is not a perfect cube because there is no number which, when multiplied 3 times with itself, gives 121 as the product.

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 3 = 3³ = 27. Here, both 3 and 27 are odd numbers.

What is an Easy Way to Calculate the 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 Formula of Cube Root?

The formula of cube root is \(a=\sqrt[3]{b}\), where a is the cube root of b. For example, the cube root of 125 is 5 because 5 × 5 × 5 = 125.