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

Step 1: Write what we need.
We want $x^{-3} + y^{-3}$. This is the same as \[ x^{-3} + y^{-3} = \frac{x^3 + y^3}{(xy)^3} \]

Step 2: Find the sum $x + y$.
With $x = 2 + \sqrt3$ and $y = 2 - \sqrt3$, \[ x + y = (2+\sqrt3) + (2-\sqrt3) = 4 \]

Step 3: Find the product $xy$.
\[ xy = (2+\sqrt3)(2-\sqrt3) = 4 - 3 = 1 \]

Step 4: Use the cube-sum identity.
\[ x^3 + y^3 = (x+y)^3 - 3xy(x+y) \]

Step 5: Plug in the values.
\[ x^3 + y^3 = 4^3 - 3(1)(4) = 64 - 12 = 52 \]

Step 6: Finish the calculation.
Since $xy = 1$, we have $(xy)^3 = 1$. \[ x^{-3} + y^{-3} = \frac{52}{1} = 52 \] \[ \boxed{52} \]