To simplify the expression \( x^{\frac{1}{9}} \times x^{\frac{1}{7}} \), we can use the property of exponents that states:
\[ x^a \times x^b = x^{a+b} \]
In this case, we have:
\[ a = \frac{1}{9} \quad \text{and} \quad b = \frac{1}{7} \]
So we need to add these two fractions:
\[ \frac{1}{9} + \frac{1}{7} \]
To add these fractions, we need a common denominator. The least common multiple of 9 and 7 is 63. We convert both fractions:
\[ \frac{1}{9} = \frac{7}{63} \] \[ \frac{1}{7} = \frac{9}{63} \]
Now, we can add them:
\[ \frac{7}{63} + \frac{9}{63} = \frac{16}{63} \]
Now we can write the expression:
\[ x^{\frac{1}{9}} \times x^{\frac{1}{7}} = x^{\frac{16}{63}} \]
Thus, the simplified expression is:
\[ \boxed{x^{\frac{16}{63}}} \]