Simplify to lowest terms, if possible: \( \frac{21 t^{3}}{49 t^{4}} \)

To simplify the expression \( \frac{21 t^{3}}{49 t^{4}} \), we begin by dividing both the numerator and the denominator by their greatest common factor. The coefficients 21 and 49 share a common factor of 7. Thus, we can simplify: \[ \frac{21}{49} = \frac{21 \div 7}{49 \div 7} = \frac{3}{7} \] Next, we simplify the \( t \) terms. Since \( t^{3} \) in the numerator and \( t^{4} \) in the denominator share the base \( t \), we can apply the exponent rule \( \frac{t^m}{t^n} = t^{m-n} \): \[ \frac{t^{3}}{t^{4}} = t^{3-4} = t^{-1} = \frac{1}{t} \] Combining these results, we have: \[ \frac{21 t^{3}}{49 t^{4}} = \frac{3}{7} \cdot \frac{1}{t} = \frac{3}{7t} \] Therefore, the simplified expression is: \[ \frac{3}{7t} \]