Question

If \(x=2+\sqrt3,\ y=2-\sqrt3\), then the value of \(x^{-3}+y^{-3}\) is

• A. 50
• B. 52
• C. 45
• D. 56

Hint:

For conjugate pairs such as \(2+\sqrt3\) and \(2-\sqrt3\), the product is often 1, making reciprocal expressions very easy to simplify.

Answer 1

The Correct Option is B

Concept: We use the identities \[ x^{-3}+y^{-3} = \frac{x^3+y^3}{(xy)^3} \] and \[ x^3+y^3=(x+y)^3-3xy(x+y) \]

Step 1: Find \(x+y\) and \(xy\).
Given \[ x=2+\sqrt3,\qquad y=2-\sqrt3 \] Adding, \[ x+y=(2+\sqrt3)+(2-\sqrt3)=4 \] Multiplying, \[ xy=(2+\sqrt3)(2-\sqrt3) \] \[ =4-3=1 \]

Step 2: Calculate \(x^3+y^3\).
Using \[ x^3+y^3=(x+y)^3-3xy(x+y) \] \[ =(4)^3-3(1)(4) \] \[ =64-12 \] \[ =52 \]

Step 3: Evaluate \(x^{-3}+y^{-3}\).
\[ x^{-3}+y^{-3} = \frac{x^3+y^3}{(xy)^3} \] \[ = \frac{52}{1^3} \] \[ =52 \] 52