To solve the problem, we first need to multiply the coefficients and then the variables. The coefficients are \frac{1}{2} and \frac{1}{3}. When we multiply these, we get \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}. Next, we look at the variable part, which is x raised to the powers of 5 and 7. When multiplying variables with the same base, we add their exponents. Therefore, x^{5} \times x^{7} = x^{5+7} = x^{12}. Combining both parts, we arrive at the final simplified expression: \frac{1}{6} x^{12}.