Which monomial is a perfect cube among the given expressions 49p9q3r24, 81p12q15r12, 121p9q3r6, and 343p6q21r6? To determine this, we need to understand the concept of a perfect cube and how it applies to monomials. A perfect cube is a number that can be expressed as the cube of an integer. In the context of monomials, this means that each term within the monomial must be a perfect cube of a variable or a constant.
Let’s analyze each monomial to see if it is a perfect cube. The first monomial, 49p9q3r24, can be broken down into its prime factors. 49 is a perfect cube (7^3), but the other terms, p9, q3, and r24, are not perfect cubes. Therefore, 49p9q3r24 is not a perfect cube.
The second monomial, 81p12q15r12, also requires prime factorization. 81 is a perfect cube (3^4), but the other terms, p12, q15, and r12, are not perfect cubes. Hence, 81p12q15r12 is not a perfect cube either.
Next, we examine the monomial 121p9q3r6. 121 is a perfect cube (11^3), but the other terms, p9, q3, and r6, are not perfect cubes. Therefore, 121p9q3r6 is not a perfect cube.
Finally, we look at the monomial 343p6q21r6. 343 is a perfect cube (7^3), and the other terms, p6, q21, and r6, are also perfect cubes (p^2, q^3, and r^2, respectively). Thus, 343p6q21r6 is a perfect cube.
In conclusion, among the given monomials, 343p6q21r6 is the only one that is a perfect cube. This is because all the terms within the monomial are perfect cubes of their respective variables or constants. Understanding the concept of perfect cubes and how they apply to monomials is essential in various mathematical contexts, such as algebra and number theory.
