How to Find the Cube Root of 1728?
The roots of a number are a very important topic in Mathematics. They form the basis of a large proportion of trigonometric and arithmetic calculations at higher-level mathematics. The root of a number is simply another number which if multiplied a certain time to itself will give the same number in return. For ex- 2 are the square root of 4 and 2 is also the cube root of 8. That is because if we multiply 2 by itself once we get 4 so we say the square of 2 is 4 and if we multiply 2 by 2 and 2 again then we get 8, which is said as 8 is the cube of 2. This can also be thought from the perspective of dimensions where every time a number is multiplied by itself, a new dimension is added. This same theory is used in the calculation and nomenclature of vector quantities in physics like velocity and acceleration.
How to Find the Cube Root of 1728
There are two methods to find the cube root of a number. They are:
The Estimation Method.
The Prime Factorization Method.
In this article, we will show the methods to find the cube root of 1728.
To calculate the cube root of 1728 by estimation method, it is important for us to learn the cubes of natural numbers from 1 to 9. These values are easy to learn and help the students to find the cube roots of any number and must be remembered by heart.
Let us now find the cube root of 1728 by following the below steps.
Consider the unit digit of 1728
The unit digit of 1728 is 8.
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.
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:
Calculate the prime factors of 1728
1728 = 2 × 2 ×2 × 2 × 2 × 2 ×3 × 3 × 3
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
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
Find the cube root of 175616 by estimation method.
Solution:
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.)
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.
Find the cube root of 10648 by the prime factorization method.
Solution:
We will initially find the prime factors of 10648
10648 = 2 × 2× 2 × 11 × 11 ×11
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
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.
FAQs on Cube Root of 1728
1. Explain the perfect cube and non-perfect cube.
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?
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.
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.
In this step, we will divide 150 by 52, i.e., 150/25 = 6.
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.
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.
Hence, the cube root of 150 is \[\sqrt[3]{150}\] = 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.
3. What is the difference between prime factorization and roots of a number?
The prime factorization of a number is the process by which a number is divided into its basic constituents or prime numbers of which it is a multiple of, whereas while finding out the root of a particular number, you have to find out which number multiplied by itself a certain number of times will give back the same number. There are various similarities in both these methods as in both the processes you have to start by factorization.
4. What do roots and cubes help in calculating area and volume?
As described above, the concept of squares and cubes can be thought of as adding a dimension into a quantity by which we can calculate the various functions of an entity. It should not be surprising that the area of a square is the square of an edge of the square while the volume of a cube is the cube of the edge of a cube. This is so because the area and volume work on the fundamentals of dimensions and hence the concepts of roots and square roots must be understood properly.
5. Explain what is a perfect cube and a non-perfect cube?
A perfect cube is a number that can be expressed as the product of multiplying an integer thrice by itself and the end number is known as the perfect cube-like 27 is a perfect cube because if 3 is multiplied thrice i.e. 3 x 3 x 3 =27. A non-perfect cube is a number that cannot be expressed as a perfect product of multiplying an integer thrice by itself. For ex- 150 cannot be expressed as a perfect cube.
6. How can the study materials along with exercise questions for cube roots be downloaded from Vedantu?
The Online resources at Vedantu can easily be accessed using 4 steps:
Open the website of vedantu on your Laptop or you can log in to the vedantu app through your phone.
Search the topic of cube roots along with the particular exercise you are looking for.
Click on Download PDF to download the solution in PDF format.
Enter OTP and then the solutions will be sent to your email ID.
7. I am having trouble while solving questions on cubes and cube roots, what shall I do?
Cube roots are very important for trigonometric and arithmetic equations of higher standards, so the students must have good command over the topic as questions regarding this shape come regularly in board examinations and also in engineering entrance examinations like the JEE Mains and JEE Advanced. Try asking your doubts to your teachers when the subject is being taught in the class. You can access the online study materials at Vedantu which are designed in a student-friendly way to ensure a proper understanding of the basics of the topic and the exercises have been provided with step-by-step solutions.
