Express `((4+sqrt15)^(3//2)+(4-sqrt15)^(3//2))/((6+sqrt35)^(3//2)-(6-sqrt35)^(3//2))` in rational form.

Answer

To express the given expression \(\frac{(4+\sqrt{15})^{3/2}+(4-\sqrt{15})^{3/2}}{(6+\sqrt{35})^{3/2}-(6-\sqrt{35})^{3/2}}\) in rational form, we will follow these steps:

### Step 1: Simplify the Numerator
The numerator is \((4+\sqrt{15})^{3/2} + (4-\sqrt{15})^{3/2}\). We can use the identity for the sum of cubes:
\[ A^3 + B^3 = (A + B)(A^2 - AB + B^2) \]
Let \(A = (4+\sqrt{15})^{1/2}\) and \(B = (4-\sqrt{15})^{1/2}\). Then we have:
\[ A^3 + B^3 = (A + B)((A^2 + B^2) - AB) \]

### Step 2: Calculate \(A + B\)
\[ A + B = (4+\sqrt{15})^{1/2} + (4-\sqrt{15})^{1/2} \]
This can be simplified by rationalizing:
\[ A + B = \sqrt{4+\sqrt{15}} + \sqrt{4-\sqrt{15}} \]

### Step 3: Calculate \(A^2 + B^2\)
\[ A^2 + B^2 = (4+\sqrt{15}) + (4-\sqrt{15}) = 8 \]

### Step 4: Calculate \(AB\)
\[ AB = \sqrt{(4+\sqrt{15})(4-\sqrt{15})} = \sqrt{16 - 15} = \sqrt{1} = 1 \]

### Step 5: Substitute Back into the Numerator
Now substituting back:
\[ A^3 + B^3 = (A + B)(A^2 + B^2 - AB) = (A + B)(8 - 1) = (A + B)(7) \]
Thus, the numerator simplifies to:
\[ 7(A + B) \]

### Step 6: Simplify the Denominator
The denominator is \((6+\sqrt{35})^{3/2} - (6-\sqrt{35})^{3/2}\). We can use the identity for the difference of cubes:
\[ A^3 - B^3 = (A - B)(A^2 + AB + B^2) \]
Let \(A = (6+\sqrt{35})^{1/2}\) and \(B = (6-\sqrt{35})^{1/2}\).

### Step 7: Calculate \(A - B\)
\[ A - B = (6+\sqrt{35})^{1/2} - (6-\sqrt{35})^{1/2} \]

### Step 8: Calculate \(A^2 + B^2\) and \(AB\)
\[ A^2 + B^2 = (6+\sqrt{35}) + (6-\sqrt{35}) = 12 \]
\[ AB = \sqrt{(6+\sqrt{35})(6-\sqrt{35})} = \sqrt{36 - 35} = \sqrt{1} = 1 \]

### Step 9: Substitute Back into the Denominator
Now substituting back:
\[ A^3 - B^3 = (A - B)(A^2 + AB + B^2) = (A - B)(12 + 1) = (A - B)(13) \]
Thus, the denominator simplifies to:
\[ 13(A - B) \]

### Step 10: Final Expression
Now we can write the entire expression as:
\[ \frac{7(A + B)}{13(A - B)} \]

### Step 11: Rational Form
Thus, the expression in rational form is:
\[ \frac{7 \left( (4+\sqrt{15})^{1/2} + (4-\sqrt{15})^{1/2} \right)}{13 \left( (6+\sqrt{35})^{1/2} - (6-\sqrt{35})^{1/2} \right)} \]