Show that 1728 is a perfect cube also find the number whose cube is -- 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.

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.

1729 is the natural number following 1728 and preceding 1730. It is a taxicab number, and is variously known as Ramanujan's number and the Ramanujan-Hardy number, after an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital. He related their conversation:[1][2][3][4]

I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

The two different ways are:

The quotation is sometimes expressed using the term "positive cubes", since allowing negative perfect cubes (the cube of a negative integer) gives the smallest solution as 91 (which is a divisor of 1729; 19 × 91 = 1729).

Numbers that are the smallest number that can be expressed as the sum of two cubes in n distinct ways[5] have been dubbed "taxicab numbers". The number was also found in one of Ramanujan's notebooks dated years before the incident, and was noted by Frénicle de Bessy in 1657. A commemorative plaque now appears at the site of the Ramanujan-Hardy incident, at 2 Colinette Road in Putney.[6]

The same expression defines 1729 as the first in the sequence of "Fermat near misses" (sequence A050794 in the OEIS) defined, in reference to Fermat's Last Theorem, as numbers of the form 1 + z3 which are also expressible as the sum of two other cubes.

Other properties[edit]

1729 is also the third Carmichael number, the first Chernick–Carmichael number (sequence A033502 in the OEIS), and the first absolute Euler pseudoprime. It is also a sphenic number.

1729 is also the third Zeisel number.[7] It is a centered cube number,[8] as well as a dodecagonal number,[9] a 24-gonal[10] and 84-gonal number.

Investigating pairs of distinct integer-valued quadratic forms that represent every integer the same number of times, Schiemann found that such quadratic forms must be in four or more variables, and the least possible discriminant of a four-variable pair is 1729.[11]

1729 is the lowest number which can be represented by a Loeschian quadratic form a2+ab+b2{\displaystyle a^{2}+ab+b^{2}}

1729 is the dimension of the Fourier transform on which the fastest known algorithm for multiplying two numbers is based.[13] This is an example of a galactic algorithm.

Is 1728 a perfect cube answer?

Is 1728 a perfect number?

Which of the following number is a perfect cube 1728 1729 1727?

Which of the following are the cube of even number 1728?