A perfect cube is an integer whose cube root is an integer (for example, 27, 64, and 125). If p and q are perfect cubes, which of the following expressions will not necessarily be a perfect cube?

Introduction / Context:
This conceptual problem explores how operations on perfect cubes behave. We are told that p and q are perfect cubes and asked to identify which constructed expression is not guaranteed to be a perfect cube. Understanding how multiplication, scalar factors, addition, and negation affect perfect cubes is important in algebra and number theory.

Given Data / Assumptions:

• p and q are perfect cubes, meaning there exist integers a and b such that p = a^3 and q = b^3.
• We are given several expressions involving p and q: 8p, pq, pq + 27, -p, and p^2q.
• We must determine which expression is not necessarily a perfect cube for all choices of p and q.
• We work with integer arithmetic and standard definitions of perfect cubes.

Concept / Approach:
We rewrite each expression using p = a^3 and q = b^3. Then we check whether each expression can be written as the cube of an integer. Multiplying perfect cubes, multiplying by another perfect cube such as 8 (which is 2^3), or taking the negative of a perfect cube all preserve the cube property. However, adding two cubes does not in general produce another cube. Carefully checking each option leads us to the correct conclusion.

Step-by-Step Solution:
Step 1: Let p = a^3 and q = b^3 for some integers a and b.Step 2: Consider option 8p. Since 8 = 2^3, we have 8p = 2^3 * a^3 = (2a)^3, which is a perfect cube.Step 3: Consider option pq. Here pq = a^3 * b^3 = (ab)^3, which is also a perfect cube.Step 4: Consider option -p. This is -a^3, which can be written as (-a)^3, so -p is still a perfect cube (just negative).Step 5: Consider option p^2q. We have p^2q = (a^3)^2 * b^3 = a^6 * b^3 = (a^2)^3 * b^3 = (a^2 b)^3, which is a perfect cube.Step 6: Now consider option pq + 27. Here pq = a^3 b^3 = (ab)^3, and 27 = 3^3. So pq + 27 = (ab)^3 + 3^3.Step 7: In general, the sum of two cubes is not itself a perfect cube. For example, take a = 1 and b = 1, so pq = 1^3 * 1^3 = 1 and pq + 27 = 1 + 27 = 28, which is not a perfect cube.Step 8: Therefore pq + 27 is not guaranteed to be a perfect cube for all choices of perfect cubes p and q.

Verification / Alternative check:
Using concrete values makes the reasoning very clear. As above, choose p = 1 (so a = 1) and q = 1 (so b = 1). Then 8p = 8, which equals 2^3 and is a perfect cube. pq = 1, which is 1^3, also a perfect cube. -p = -1, which is (-1)^3, a perfect cube. p^2q = 1^2 * 1 = 1, again a cube. But pq + 27 = 1 + 27 = 28. There is no integer k such that k^3 = 28, because 3^3 = 27 and 4^3 = 64. Hence 28 is not a perfect cube, showing that pq + 27 fails in this simple case.

Why Other Options Are Wrong:
Option 8p: Because 8 is 2^3, multiplying p by 8 simply forms the cube of 2a.Option pq: The product of two perfect cubes is always a perfect cube, as it equals (ab)^3.Option -p: The negative of a perfect cube is still a perfect cube of a negative integer.Option p^2q: This equals (a^2 b)^3, which is again a perfect cube.

Common Pitfalls:
Students sometimes think that the sum of two perfect cubes is another perfect cube, which is almost never true except for special cases. Another error is to assume that negative numbers cannot be perfect cubes, but in integer arithmetic, cubes of negative integers are negative perfect cubes. Always express the given cubes in terms of a^3 and b^3, and then algebraically manipulate each option to see whether it remains the cube of an integer.

Final Answer:
The expression that is not necessarily a perfect cube is pq + 27.