Certainly! Let's solve for \( x + y + xy \) given that \( x = \frac{1}{3} + 2\sqrt{2} \) and \( y = \frac{1}{3} + 2\sqrt{2} \).
Step 1: Calculate \( x + y \)
- Since both \( x \) and \( y \) are equal, we can simplify:
\[
x + y = x + x = 2x
\]
- Substitute the value of \( x \):
\[
x + y = 2\left(\frac{1}{3} + 2\sqrt{2}\right) = \frac{2}{3} + 4\sqrt{2}
\]
Step 2: Calculate \( xy \)
- Since \( x = y \), we have:
\[
xy = x \cdot x = x^2
\]
- Now, substitute the value of \( x \):
\[
x^2 = \left(\frac{1}{3} + 2\sqrt{2}\right)^2
\]
- Expanding this expression using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \):
\[
x^2 = \left(\frac{1}{3}\right)^2 + 2\left(\frac{1}{3}\right)(2\sqrt{2}) + (2\sqrt{2})^2
\]
\[
= \frac{1}{9} + \frac{4\sqrt{2}}{3} + 8
\]
- Combining the terms gives:
\[
= \frac{1}{9} + \frac{4\sqrt{2}}{3} + \frac{72}{9} = \frac{73}{9} + \frac{4\sqrt{2}}{3}
\]
Step 3: Combine \( x + y + xy \)
- Now, substitute back into our original expression:
\[
x + y + xy = \left(\frac{2}{3} + 4\sqrt{2}\right) + \left(\frac{73}{9} + \frac{4\sqrt{2}}{3}\right)
\]
- To combine, find a common denominator (which is 9):
\[
= \frac{6}{9} + 4\sqrt{2} + \frac{73}{9} + \frac{12\sqrt{2}}{9}
\]
- Combine like terms:
\[
= \frac{79}{9} + \frac{36\sqrt{2}}{9} = \frac{79 + 36\sqrt{2}}{9}
\]
Final Answer
- Therefore, the value of \( x + y + xy \) is:
\[
\boxed{\frac{79 + 36\sqrt{2}}{9}}
\]