Which monomial is a perfect cube among 16×6, 27×8, 32×12, and 64×6? This question often arises in mathematics, especially when dealing with algebraic expressions and cube roots. A perfect cube is a number that can be expressed as the cube of an integer. In this article, we will explore the given monomials and determine which one is a perfect cube.
In mathematics, a monomial is a product of a constant and one or more variables, each raised to a non-negative integer power. A perfect cube monomial, therefore, is a monomial where the variables are raised to powers that are multiples of three. Let’s analyze each of the given monomials to see if they fit this criterion.
The first monomial is 16×6. To determine if it is a perfect cube, we need to check if the exponent of x, which is 6, is a multiple of 3. Since 6 is divisible by 3, we can proceed to check if the constant term, 16, is a perfect cube. The cube root of 16 is 2.5198421, which is not an integer. Therefore, 16×6 is not a perfect cube.
The second monomial is 27×8. Here, the exponent of x, 8, is also a multiple of 3. To check if 27 is a perfect cube, we need to find its cube root. The cube root of 27 is 3, which is an integer. Thus, 27×8 is a perfect cube.
Moving on to the third monomial, 32×12. The exponent of x, 12, is not a multiple of 3. Therefore, we can conclude that 32×12 is not a perfect cube.
Finally, we have the monomial 64×6. The exponent of x, 6, is a multiple of 3. However, the cube root of 64 is 4, which is an integer. Thus, 64×6 is a perfect cube.
In conclusion, among the given monomials, 27×8 and 64×6 are perfect cubes. These monomials have variables raised to powers that are multiples of three, and their constant terms are perfect cubes. Understanding the concept of perfect cubes and their relationship with monomials is crucial in various mathematical applications, such as factoring, simplifying expressions, and solving equations.
