Find the third proportional to the expressions (x − y)^2 and (x^2 − y^2)^2 in algebraic form.

Introduction / Context:
This is an algebraic ratio and proportion problem involving expressions rather than simple numbers. We are asked to find the third proportional to two given expressions. This tests understanding of the definition of third proportional and the ability to manipulate algebraic expressions using identities such as x^2 − y^2 = (x − y)(x + y).

Given Data / Assumptions:

• The first expression is (x − y)^2.• The second expression is (x^2 − y^2)^2.• We must find an expression Z that is third proportional to these two, so that (x − y)^2 : (x^2 − y^2)^2 = (x^2 − y^2)^2 : Z.• All expressions are defined for values of x and y that make denominators nonzero if any arise.

Concept / Approach:
For numbers or expressions a and b, a third proportional c is defined so that a : b = b : c. In algebraic terms, this means a / b = b / c, leading to a * c = b^2. Solving for c gives c = b^2 / a. Here a = (x − y)^2 and b = (x^2 − y^2)^2. So Z = ((x^2 − y^2)^2)^2 / (x − y)^2. To simplify this, we use the factorization x^2 − y^2 = (x − y)(x + y). Substituting this into the expression and simplifying powers allows us to express Z in terms of (x − y) and (x + y).

Step-by-Step Solution:
Step 1: Let a = (x − y)^2 and b = (x^2 − y^2)^2.Step 2: The third proportional Z must satisfy a : b = b : Z.Step 3: Using the relation for third proportional, a * Z = b^2, so Z = b^2 / a.Step 4: Substitute a and b: Z = ((x^2 − y^2)^2)^2 / (x − y)^2.Step 5: Simplify ((x^2 − y^2)^2)^2 as (x^2 − y^2)^4.Step 6: So Z = (x^2 − y^2)^4 / (x − y)^2.Step 7: Use the identity x^2 − y^2 = (x − y)(x + y).Step 8: Then (x^2 − y^2)^4 = ((x − y)(x + y))^4 = (x − y)^4 * (x + y)^4.Step 9: Substitute back: Z = (x − y)^4 * (x + y)^4 / (x − y)^2.Step 10: Simplify by canceling powers of (x − y): (x − y)^4 / (x − y)^2 = (x − y)^2.Step 11: Therefore, Z = (x − y)^2 * (x + y)^4.Step 12: The simplified expression is (x + y)^4 (x − y)^2.

Verification / Alternative check:
To verify, we check whether the proportion holds: (x − y)^2 : (x^2 − y^2)^2 = (x^2 − y^2)^2 : (x + y)^4 (x − y)^2. Writing this as fractions, we want (x − y)^2 / (x^2 − y^2)^2 = (x^2 − y^2)^2 / ((x + y)^4 (x − y)^2). Using x^2 − y^2 = (x − y)(x + y), the left side becomes (x − y)^2 / ((x − y)^2 (x + y)^2) = 1 / (x + y)^2. The right side becomes ((x − y)^2 (x + y)^2) / ((x + y)^4 (x − y)^2) = 1 / (x + y)^2. Both sides match, confirming the correctness of the third proportional.

Why Other Options Are Wrong:
• (x + y)^3 (x − y)^2: This has the wrong power of (x + y) and does not satisfy the proportion when checked.• (x + y)^2 (x − y)^2: This corresponds to b, not to the third proportional b^2 / a.• (x + y)^2 (x − y)^3: This has an extra power of (x − y) and does not simplify correctly in the proportion test.

Common Pitfalls:
Many learners misapply the definition of third proportional and instead compute something like a * b or a + b. Others may mis-handle the algebraic identity x^2 − y^2 = (x − y)(x + y) or incorrectly raise expressions to powers. Canceling powers incorrectly or missing a factor in the numerator or denominator when simplifying is also common. Being systematic with the definition Z = b^2 / a and carefully using standard algebraic identities prevents these mistakes.

Final Answer:
The third proportional to (x − y)^2 and (x^2 − y^2)^2 is (x + y)^4 (x − y)^2.