If `a= (sqrt3 - sqrt2)/(sqrt3 + sqrt2), b = (sqrt3 + sqrt2)/(sqrt3 - sqrt2)` then what is the value of `a^2/b+b^2/a` ?

Options

A. 970
B. 1030
C. 930
D. 900

Answer

To solve the problem, we need to find the value of \( \frac{a^2}{b} + \frac{b^2}{a} \) given:

\[ a = \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}, \quad b = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}}. \]

### Step 1: Rationalize \( a \)

To rationalize \( a \), we multiply the numerator and denominator by the conjugate of the denominator:

\[ a = \frac{(\sqrt{3} - \sqrt{2})(\sqrt{3} - \sqrt{2})}{(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})}. \]

Calculating the denominator:

\[ (\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2}) = 3 - 2 = 1. \]

Calculating the numerator:

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

Thus,

\[ a = 5 - 2\sqrt{6}. \]

### Step 2: Rationalize \( b \)

Similarly, we rationalize \( b \):

\[ b = \frac{(\sqrt{3} + \sqrt{2})(\sqrt{3} + \sqrt{2})}{(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2})}. \]

Calculating the denominator:

\[ (\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2}) = 3 - 2 = 1. \]

Calculating the numerator:

\[ (\sqrt{3} + \sqrt{2})^2 = 3 + 2\sqrt{6} + 2 = 5 + 2\sqrt{6}. \]

Thus,

\[ b = 5 + 2\sqrt{6}. \]

### Step 3: Calculate \( a^2 \) and \( b^2 \)

Now we calculate \( a^2 \) and \( b^2 \):

\[ a^2 = (5 - 2\sqrt{6})^2 = 25 - 20\sqrt{6} + 24 = 49 - 20\sqrt{6}, \]

\[ b^2 = (5 + 2\sqrt{6})^2 = 25 + 20\sqrt{6} + 24 = 49 + 20\sqrt{6}. \]

### Step 4: Calculate \( a^2/b + b^2/a \)

Now we need to find:

\[ \frac{a^2}{b} + \frac{b^2}{a}. \]

Finding a common denominator:

\[ \frac{a^2}{b} + \frac{b^2}{a} = \frac{a^3 + b^3}{ab}. \]

### Step 5: Calculate \( ab \)

Calculating \( ab \):

\[ ab = (5 - 2\sqrt{6})(5 + 2\sqrt{6}) = 25 - (2\sqrt{6})^2 = 25 - 24 = 1. \]

### Step 6: Calculate \( a^3 + b^3 \)

Using the identity \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \):

First, find \( a + b \):

\[ a + b = (5 - 2\sqrt{6}) + (5 + 2\sqrt{6}) = 10. \]

Now find \( a^2 + b^2 \):

\[ a^2 + b^2 = (49 - 20\sqrt{6}) + (49 + 20\sqrt{6}) = 98. \]

Now substitute into \( a^3 + b^3 \):

\[ a^3 + b^3 = (10)(98 - 1) = 10 \times 97 = 970. \]

### Step 7: Final Calculation

Now we can substitute back:

\[ \frac{a^3 + b^3}{ab} = \frac{970}{1} = 970. \]

Thus, the value of \( \frac{a^2}{b} + \frac{b^2}{a} \) is:

\[ \boxed{970}. \]