Cube Root of 1728

Bookmark added to your notes.
View Notes
×

How to Find the Cube Root of 1728?

In Mathematics, the cube root is a special value. With the help of this special value, we can multiply the given number thrice. For example, 6 × 6 × 6 = 216. Hence, the cube root of 216 is 6. Similarly, the cube root of 1728 is 12. The cube root of 1728 is represented as ∛1728. It is a value that obtains the original number i.e. 1728 on multiple by itself thrice. Hence, we can say that cube root is an inverse process of calculating cube of any number.


Cube Root of 1728, ∛1728 = 12


Let us consider ∛1728 = k then 1728 = k³. Here, we will find the value of k. If we use the prime factorization method to find the cube root of 1728, it will be tedious and time- consuming. Hence, the estimation method is the easiest and quickest method to find the cube root of 1728. Estimation methods can be used to find the cubic root of equals to or more than four-digit numbers.


(image will be uploaded soon)


What is a Cube Root?

A cube root is the number that is multiplied by itself three times in order to obtain a cubic value.

∛ is generally used to represent the cube root . For example: ∛1728 = ∛(12 × 12 × 12) = 12. It is quite easy to find the cube root of 1728 as it is a perfect cube. Hence, the cube root of 1728 is 12.


Methods to FInd the Cube Root of 1728

The two methods to find the cube root of 1728  are:

  1. Estimation Method.

  2. Prime Factorization Method.

In this article, we will discuss the methods to find the cube root of 1728 .


To find the cube root of 1728  by estimation method, it is necessary for us to learn the cubes of natural numbers from 1 to 9. These values are simple to learn and help the students to find the cube roots of any number within no time.


Cube Root of 1728 by Estimation Method

Number

Cubes (n³)

1

1

2

8

3

27

4

64

5

125

6

216

7

343

8

512

9

729

10

1000


Let us now find the cube root of 1728  by following the below steps.

  1. Consider the unit digit of 1728 

The unit digit of 1728  is 8.

  1. With the help of the cube table given above, check the cube of which number has 8 at its unit place.

Clearly,  we can see = 2³ = 8

It implies that the cube root of 1728  has 2 at its unit place

So, we can say that the unit digit of the cube root of 1728  is 2.

  1. Now, ignore the last 3 digits of 1728  i.e.728.

Considering 1 as a benchmark digit, we can see the cube of 1 is equal to 1.

Therefore, we obtain the cube root of 1728 in two-digit.

Hence, the cube root of 1728 is 12.


Cube Root of 1728 by Prime Factorization Method

Now, we will learn to find the cube root of 1728 by the prime factorization method. In the prime factorization method, we will first find the prime factors of 1728. After finding the prime factors of 1728, we will pair similar factors in a group of 3 to denote them as cubes. We will get the required value because cubes of a number ignore the cube roots.


Let us learn to find the cube root of 1728 through the prime factorization method step by step:

  1. Calculate the prime factors of 1728

1728 = 2 × 2 ×2 × 2 × 2 × 2 ×3 × 3 × 3

  1. Pair the similar factors in a group of them and represent them as cubes.

1728 =  (2 × 2 × 2) × (2 × 2 × 2 ) × (3 × 3 ×3)

1728 = 2³ × 2³ × 3³

  1. Apply cube root on both the left and right side of the above expression.

\[\sqrt[3]{1728} = \sqrt[3]{2^{3} \times 2^{3} \times 3^{3}} = 2 \times 2 \times 3 = 12\].

The cube root gets neutralized by the cube of 12.

Hence, the cube root of 1728 is 12.


Solved Example

1. Find the cube root of 175616 by estimation method.


Solution:

  1. Consider the last 3 digits of 175616 as the first half and the remaining digit as the second half

The first part of 175616 is 616 and the second part is 175.

Now, look at the last 3 digits of 175616 and with the help of the cubes table given above find the cube of a digit (from 0 to 9) that has the last digit 6.

So, the unit place of the cube root of 175616 is 6.

(6³ = 216, the last digit of 216 is 6.)

  1. The second part of a given number is 175.

175 lies in between the cubes of 5 and 6 ( i.e. in between the 125 and 216).

Take the lowest number among the two given numbers 5 and 6. The lowest number here is 5

Hence, the tenth digit of the cube root of 175616 is 5.

Therefore, the cube root of 175616 is 56.


2. Find the cube root of 10648 by the prime factorization method.


Solution:

  1. We will initially find the prime factors of 10648

       10648 = 2 × 2× 2 × 11 × 11 ×11

  1. We will pair the factors in a group and represent them as cubes.

10648 =  (2 × 2× 2) × (11 × 11 × 11)

10648= 2³ × 11³ [ By exponent law ab + ac = ab+c ]

10648 = (2 × 11)³  [ By exponent law ab + ac = ab+c ]

10648 = 22³

  1. Apply cube root on both the left and right side of the above expression.

\[\sqrt[3]{10648} = \sqrt[3]{22^{3}}\]

The cube root gets neutralized by the cube of 22.

Hence, the cube root of 10648 is 22.


Quiz Time

1. What is the one digit of the cube of 11?

  1. 1

  2. 2

  3. 3

  4. 4

2. Which of the numbers given below is not a perfect cube?

  1. 216

  2. 1000

  3. 243

  4. 1331

FAQ (Frequently Asked Questions)

1. Explain the Perfect Cube and Non-Perfect Cube.

Ans. Perfect Cube - A perfect cube is a number that can be represented as the product of three equal integers. In other words, perfect cubes are those numbers that are formed by cubing an integer. For example , 27 is a perfect cube  because 3³ = 3 × 3 × 3 = 27.


Non-Perfect Cube - A non- perfect cube is a number that cannot be written as the product of three equal integers. In other words, perfect cubes are those numbers that cannot be formed by cubing an integer. For example, 150 is a non- perfect cube as it cannot be formed by cubing any integer.

2. How to Find the Cube Root of Non- Perfect Cubes?

Ans. There are multiple numbers that are not perfect cubes and we cannot determine the cube root of such numbers using the prime factorization and estimation method. Hence, we will use some helping tricks to calculate the cube root of non- perfect cubes.


Let us calculate the cube root of 150 which is a non -perfect cube step by step.

  1. The digit 150 lies between 125 (the cube of 5) and 216 (the cube of 6). So, we will consider the lowest digit here, i.e. 5.

  2. In this step, we will divide 150 by 5², i.e., 150/25 = 6.

  3. Now, we will subtract the digit 5 from 6 (whichever is greater) and divide the result by 3. So, 6 - 5 = 1 & ⅓ = 0.333.

  4. In the last step, we will add the lowest number i.e. 5 which we got in the step one and the decimal number i.e. 0.333 which we got in step 3. Hence, 5 + 0.33 = 5.33.

  5. Hence, the cube root of 150 is ∛150 = 5.3.

If we calculate the cube root of 150 through a calculator, we will get the value approximately equal to the actual value, i.e. 5.314.