If P = 7 + 4√3 and P × Q = 1, then what is the value of (1/P^2) + (1/Q^2)?

Introduction / Context:
This problem involves surds and reciprocals of expressions containing square roots. You are given a number P with a surd part and told that P multiplied by Q equals 1, which means Q is the multiplicative inverse of P. The task is to compute (1/P^2) + (1/Q^2). Problems of this type test your ability to manipulate surds, use conjugates and simplify expressions involving reciprocals.

Given Data / Assumptions:

• P = 7 + 4√3.
• P × Q = 1, so Q is equal to 1/P.
• We must find the value of (1/P^2) + (1/Q^2).
• All operations are in the real number system.

Concept / Approach:
Since P × Q = 1, Q is the reciprocal of P. That implies Q = 1/P, and consequently 1/Q = P. Squaring both sides gives 1/Q^2 = P^2. Therefore, (1/P^2) + (1/Q^2) simplifies to (1/P^2) + P^2. This reduces the problem to computing P^2 and then adding its reciprocal. To handle P^2, we can expand (7 + 4√3)^2 using the binomial formula and simplify using the fact that (√3)^2 = 3.

Step-by-Step Solution:
Step 1: From P × Q = 1, deduce Q = 1/P. Step 2: Take reciprocals: 1/Q = P, so 1/Q^2 = P^2. Step 3: The required sum becomes (1/P^2) + (1/Q^2) = (1/P^2) + P^2. Step 4: Compute P^2 = (7 + 4√3)^2 = 7^2 + 2 × 7 × 4√3 + (4√3)^2. Step 5: Evaluate each term: 7^2 = 49, 2 × 7 × 4√3 = 56√3, and (4√3)^2 = 16 × 3 = 48, so P^2 = 49 + 56√3 + 48 = 97 + 56√3. Step 6: Find the reciprocal by rationalising or noting that (7 - 4√3) is the conjugate whose product with P gives 1, so 1/P = 7 - 4√3 and 1/P^2 = (7 - 4√3)^2 = 97 - 56√3. Step 7: Add P^2 and 1/P^2: (97 + 56√3) + (97 - 56√3) = 194.

Verification / Alternative check:
An alternative way is to recognise that if P + 1/P is known, then P^2 + 1/P^2 can be computed using the identity (P + 1/P)^2 = P^2 + 2 + 1/P^2. Here, P + 1/P equals (7 + 4√3) + (7 - 4√3) = 14. So (P + 1/P)^2 = 14^2 = 196. Then P^2 + 1/P^2 = 196 - 2 = 194. This matches the previous result and confirms the correctness of the answer.

Why Other Options Are Wrong:

• Options a (148), b (189), d (204) and e (154) arise from various arithmetic mistakes, such as miscomputing the square, forgetting to cancel the surd parts, or misapplying the identity for P^2 + 1/P^2.
• None of these alternatives match the consistent results from both the direct expansion and the identity method.

Common Pitfalls:
Students often make sign errors when squaring expressions with surds or forget that the middle term in the square, 2ab, involves both coefficients and the radical. Another common issue is improper handling of reciprocals, especially when dealing with conjugates. Remember that multiplying an expression with its conjugate eliminates the square root term, making it easier to compute reciprocals and squares. Careful stepwise expansion and checking with identities reduce the chance of mistakes.

Final Answer:
The value of (1/P^2) + (1/Q^2) is 194.