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. What is the cube root of 1728 and why is it considered a perfect cube?
The cube root of 1728 is 12. A number is called a perfect cube if its cube root is a whole number. Since 12 is a whole number and 12 × 12 × 12 = 1728, the number 1728 is a perfect cube.
2. How can the prime factorisation method be used to find the cube root of 1728?
The prime factorisation method involves breaking a number down into its prime factors and grouping them in triplets. For 1728, the steps are:
- Step 1: Find the prime factors of 1728.
1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3. - Step 2: Group the identical factors into triplets.
1728 = (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3). - Step 3: Take one factor from each triplet and multiply them together.
Cube Root = 2 × 2 × 3 = 12.
This method confirms that the cube root of 1728 is 12.
3. What is the importance of understanding the difference between a perfect cube and a non-perfect cube?
Understanding the difference is crucial for problem-solving. A perfect cube, like 1728, has an integer as its cube root (12), which simplifies calculations. A non-perfect cube, like 1729, has a cube root that is an irrational number (approximately 12.0023), meaning its decimal representation goes on forever without repeating. Knowing this distinction helps in estimating answers and choosing the correct method for finding roots.
4. How does knowing the cube root of 1728 apply to real-world problems in geometry?
The concept of a cube root has a direct application in calculating dimensions. For example, if you have a cube-shaped box with a volume of 1728 cubic centimetres (cm³), finding the length of one of its sides requires calculating the cube root of its volume. The cube root of 1728 is 12, so the side length of the box would be 12 cm. This is a fundamental concept in geometry and spatial reasoning.
5. Why is the cube root of 1728 different from its square root?
The cube root and square root represent different operations. The cube root of a number is the value that, when multiplied by itself three times, equals the original number (12³ = 1728). The square root is the value that, when multiplied by itself twice, equals the number. While the cube root of 1728 is a whole number (12), its square root (√1728) is approximately 41.57, which is an irrational number. This shows that 1728 is a perfect cube but not a perfect square.
6. How can you estimate the cube root of 1728 just by observing its unit digit?
You can estimate the cube root of a perfect cube by looking at its digits. The number 1728 ends in 8. In cubes of single-digit numbers (0-9), only 2³ ends in 8 (2³=8). So, the unit digit of the cube root must be 2. Next, ignore the last three digits (728) and consider the remaining part, which is 1. The largest cube less than or equal to 1 is 1³ = 1. So, the tens digit is 1. Combining these, the estimated cube root is 12.
7. Is it possible to find the cube root of a negative number, such as -1728?
Yes, unlike square roots, the cube root of a negative number is a real, negative number. This is because multiplying a negative number by itself three times results in a negative product. For example, (-12) × (-12) × (-12) = 144 × (-12) = -1728. Therefore, the cube root of -1728 is -12. This is an important property that distinguishes odd-powered roots from even-powered roots.
