If \(x=\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}\) and \(y=\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}\), then the value of \(x^{3}+y^{3}\) is: (a) 950 (b) 730 (c) 650 (d) 970

Evaluation of \(x^3 + y^3\) Given \(x\) and \(y\)

For the given values of \(x\) and \(y\), defined as:

\[x=\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}\]

\[y=\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}\]

We aim to find the value of \(x^3 + y^3\).

Step 1: Simplify \(x\) and \(y\)

First, rationalize \(x\) by multiplying the numerator and denominator by the conjugate of the denominator:

\[
x=\frac{(\sqrt{3}-\sqrt{2})^2}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})} = \frac{3 – 2\sqrt{6} + 2}{1} = 5 – 2\sqrt{6}
\]

Using a similar process for \(y\), we rationalize by multiplying by the conjugate, yielding:

\[
y=\frac{(\sqrt{3}+\sqrt{2})^2}{(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2})} = 5 + 2\sqrt{6}
\]

Step 2: Calculate \(x+y\)

Adding \(x\) and \(y\) gives us:

\[
x+y = (5 – 2\sqrt{6}) + (5 + 2\sqrt{6}) = 10
\]

Step 3: Calculate \(xy\)

Multiplying \(x\) and \(y\) provides:

\[
xy = (5 – 2\sqrt{6})(5 + 2\sqrt{6}) = 25 – (2\sqrt{6})^2 = 25 – 24 = 1
\]

Step 4: Derive \(x^3 + y^3\)

Using the identity \(x^3 + y^3 = (x + y)^3 – 3xy(x + y)\) and substituting the calculated values:

\[
x^3 + y^3 = 10^3 – 3 \cdot 1 \cdot 10 = 1000 – 30 = 970
\]

Conclusion

The value of \(x^3 + y^3\) is 970, corresponding to:

(d) 970