Express with rational denominator :
`(sqrt(8)+root3(4))/(sqrt((8))-root3((4)))`

Answer

To express the given expression \((\sqrt{8} + \sqrt[3]{4}) / (\sqrt{8} - \sqrt[3]{4})\) with a rational denominator, we will follow these steps:

### Step 1: Rewrite the Roots in Exponential Form
First, we can express the square root and cube root in terms of powers of 2:
- \(\sqrt{8} = 8^{1/2} = (2^3)^{1/2} = 2^{3/2}\)
- \(\sqrt[3]{4} = 4^{1/3} = (2^2)^{1/3} = 2^{2/3}\)

Thus, we can rewrite the expression as:
\[ \frac{2^{3/2} + 2^{2/3}}{2^{3/2} - 2^{2/3}} \]

### Step 2: Rationalize the Denominator
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator:
\[ \frac{(2^{3/2} + 2^{2/3})(2^{3/2} + 2^{2/3})}{(2^{3/2} - 2^{2/3})(2^{3/2} + 2^{2/3})} \]

### Step 3: Apply the Difference of Squares Formula
Using the difference of squares formula \(a^2 - b^2 = (a - b)(a + b)\), we can simplify the denominator:
- \(a = 2^{3/2}\)
- \(b = 2^{2/3}\)

Thus, the denominator becomes:
\[ (2^{3/2})^2 - (2^{2/3})^2 = 2^{3} - 2^{4/3} \]

### Step 4: Expand the Numerator
Now, expanding the numerator:
\[ (2^{3/2} + 2^{2/3})^2 = (2^{3/2})^2 + 2 \cdot 2^{3/2} \cdot 2^{2/3} + (2^{2/3})^2 \]
Calculating each term:
- \((2^{3/2})^2 = 2^{3} = 8\)
- \((2^{2/3})^2 = 2^{4/3}\)
- \(2 \cdot 2^{3/2} \cdot 2^{2/3} = 2^{3/2 + 2/3 + 1} = 2^{3/2 + 2/3 + 3/3} = 2^{13/6}\)

So, the numerator becomes:
\[ 8 + 2^{13/6} + 2^{4/3} \]

### Step 5: Combine the Results
Now we have:
\[ \frac{8 + 2^{13/6} + 2^{4/3}}{2^{3} - 2^{4/3}} \]

### Step 6: Simplify the Denominator
The denominator can be simplified further:
\[ 2^{3} - 2^{4/3} = 2^{9/3} - 2^{4/3} = 2^{4/3}(2^{5/3} - 1) \]

### Final Expression
Now, substituting back into our expression, we get:
\[ \frac{8 + 2^{13/6} + 2^{4/3}}{2^{4/3}(2^{5/3} - 1)} \]

### Conclusion
Thus, the expression \((\sqrt{8} + \sqrt[3]{4}) / (\sqrt{8} - \sqrt[3]{4})\) with a rational denominator is:
\[ \frac{8 + 2^{13/6} + 2^{4/3}}{2^{4/3}(2^{5/3} - 1)} \]