To solve the equation \( \frac{9}{4}w = \frac{13}{14} + 2 \), we first need to simplify the right side of the equation.
Start by converting the whole number 2 to a fraction with a common denominator of 14:
\[ 2 = \frac{2 \times 14}{1 \times 14} = \frac{28}{14} \]
Now, we can combine the fractions on the right side:
\[ \frac{13}{14} + \frac{28}{14} = \frac{13 + 28}{14} = \frac{41}{14} \]
Now the equation is:
\[ \frac{9}{4}w = \frac{41}{14} \]
Next, we want to isolate \( w \). We can do this by multiplying both sides by the reciprocal of \( \frac{9}{4} \), which is \( \frac{4}{9} \):
\[ w = \frac{41}{14} \cdot \frac{4}{9} \]
Now we multiply the fractions:
\[ w = \frac{41 \cdot 4}{14 \cdot 9} = \frac{164}{126} \]
Next, we simplify \( \frac{164}{126} \) by finding the greatest common divisor (GCD) of 164 and 126. The GCD is 2, so we divide both the numerator and the denominator by 2:
\[ w = \frac{164 \div 2}{126 \div 2} = \frac{82}{63} \]
Thus, the solution for \( w \) is:
\[ \boxed{\frac{82}{63}} \]