Is 2744 a perfect cube? This question often arises when dealing with numbers and their properties. In this article, we will explore the concept of perfect cubes, determine whether 2744 is a perfect cube, and discuss the significance of this number in mathematics.
A perfect cube is a number that can be expressed as the cube of an integer. In other words, if a number \( n \) is a perfect cube, then there exists an integer \( x \) such that \( n = x^3 \). For example, 8 is a perfect cube because \( 2^3 = 8 \), and 27 is a perfect cube because \( 3^3 = 27 \).
To determine if 2744 is a perfect cube, we need to find an integer \( x \) such that \( x^3 = 2744 \). One way to do this is by using prime factorization. Prime factorization involves expressing a number as a product of its prime factors. Let’s begin by finding the prime factors of 2744.
The prime factorization of 2744 can be found by dividing the number by its smallest prime factor, which is 2. We continue dividing by 2 until we can no longer divide by 2. Then, we move on to the next smallest prime factor, which is 3, and repeat the process. This process is as follows:
2744 ÷ 2 = 1372
1372 ÷ 2 = 686
686 ÷ 2 = 343
343 ÷ 3 = 114
114 ÷ 2 = 57
57 ÷ 3 = 19
19 is a prime number.
The prime factorization of 2744 is \( 2^4 \times 3^2 \times 19 \). Now, we need to determine if we can express 2744 as a cube of an integer. To do this, we need to group the prime factors into sets of three, as each set of three will represent the cube of a prime factor.
In the prime factorization of 2744, we have two sets of three prime factors: \( 2^4 \) and \( 3^2 \). However, we have one prime factor, 19, left ungrouped. Since we cannot group 19 with any other prime factor to form a set of three, we cannot express 2744 as a cube of an integer.
Therefore, the answer to the question “Is 2744 a perfect cube?” is no. 2744 is not a perfect cube because it cannot be expressed as the cube of an integer. This highlights the importance of prime factorization in determining the properties of numbers and their relationships with other numbers.
