If √x − √y = 1 and √x + √y = 17, then what is the value of √(xy)? Solve for √x and √y first, then compute √(xy) exactly without approximation.

Introduction / Context:
This algebra question tests solving a system involving square roots by using addition and subtraction. The expressions √x − √y and √x + √y are given directly, which is ideal because you can add the equations to eliminate √y and subtract them to eliminate √x. Once you find √x and √y, you can compute √(xy) using the property √(xy) = √x * √y (for non-negative x and y). Students often make mistakes by squaring too early, mixing x with √x, or forgetting that the asked value is √(xy), not xy. The cleanest route is to treat √x and √y as variables and solve the linear system first.

Given Data / Assumptions:

• √x − √y = 1 • √x + √y = 17 • x and y are non-negative (so square roots are real) • Required: √(xy)

Concept / Approach:
Let p = √x and q = √y. Then: p − q = 1 and p + q = 17. Solve these two linear equations for p and q. Then use: √(xy) = √x * √y = p*q. This avoids unnecessary squaring and keeps the steps short and accurate.

Step-by-Step Solution:
1) Let p = √x and q = √y. 2) Then p − q = 1 and p + q = 17. 3) Add the equations: (p − q) + (p + q) = 1 + 17 ⇒ 2p = 18 ⇒ p = 9 4) Subtract the first from the second: (p + q) − (p − q) = 17 − 1 ⇒ 2q = 16 ⇒ q = 8 5) Compute √(xy): √(xy) = p*q = 9*8 = 72

Verification / Alternative check:
If √x = 9 and √y = 8, then √x − √y = 9 − 8 = 1 and √x + √y = 9 + 8 = 17, matching both conditions. Also xy = (9^2)(8^2) = 81*64 = 5184, and √5184 = 72, confirming the same result by direct computation.

Why Other Options Are Wrong:
• 24 and 32: come from adding or subtracting 9 and 8 incorrectly. • √72: confuses √(xy) with √x + √y style expressions. • 64: equals y, not √(xy).

Common Pitfalls:
• Treating √x as x and √y as y. • Squaring the equations immediately, creating extra terms and confusion. • Forgetting that √(xy) = √x * √y for non-negative x and y.

Final Answer:
72