Is 162 a perfect cube? This question often arises when exploring the properties of numbers and their cube roots. In this article, we will delve into the concept of perfect cubes and determine whether 162 fits into this category.
A perfect cube is a number that can be expressed as the cube of an integer. For example, 8 is a perfect cube because it is the cube of 2 (2^3 = 8). Similarly, 27 is a perfect cube because it is the cube of 3 (3^3 = 27). To determine if 162 is a perfect cube, we need to find an integer whose cube equals 162.
Let’s begin by taking the cube root of 162:
162^(1/3) ≈ 5.787
The cube root of 162 is approximately 5.787. Since this value is not an integer, we can conclude that 162 is not a perfect cube. In other words, there is no integer whose cube is equal to 162.
Understanding why 162 is not a perfect cube can be further explored by examining the prime factorization of the number:
162 = 2 × 81
162 = 2 × 3^4
The prime factorization of 162 consists of 2 and 3. Since the exponent of 2 is 1 and the exponent of 3 is 4, it is clear that the number cannot be expressed as the cube of an integer. For a number to be a perfect cube, all the exponents in its prime factorization must be multiples of 3.
In conclusion, 162 is not a perfect cube because its cube root is not an integer, and its prime factorization does not meet the criteria for a perfect cube. Exploring this concept can help us better understand the properties of numbers and their relationships with perfect cubes.
