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:

1. The Estimation Method.

2. 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

1. 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.

2. 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.