Is 112 a perfect cube?

Is 112 a Perfect Cube?

When exploring the properties of numbers, one common question is whether a given number is a perfect cube. In the case of 112, the answer is no — it is not a perfect cube.

To understand why, let's break it down. 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’s 2³, and 27 is 3³. To determine if 112 fits this definition, we look at its prime factorization.

The prime factorization of 112 is 2 × 2 × 2 × 2 × 7, or more clearly, 2⁴ × 7¹. For a number to be a perfect cube, all the exponents in its prime factorization must be multiples of 3. Here, the exponent of 2 is 4 (not a multiple of 3), and 7 has an exponent of 1 — which also doesn't meet the requirement.

Because the prime factors aren't grouped in threes, 112 cannot be written as the cube of a whole number. This means its cube root is not an integer, in fact, it's an irrational number. You won’t find a clean, whole number that, when multiplied by itself three times, gives 112.

So, while 112 is a composite number with several factors, it falls short of being a perfect cube. It’s a good reminder that not every number with repeated factors qualifies, the grouping must align perfectly in threes. In the world of cubes, 112 just doesn’t measure up.