Simplify. Assume \( s \) and \( t \) are greater than or equal to zero. \( \sqrt{147 s^{5} t^{4}} \)

To simplify \( \sqrt{147 s^{5} t^{4}} \), we start by breaking it down into its components. The number \( 147 \) can be factored as \( 49 \times 3 \), where \( 49 \) is a perfect square. So, we can write: \[ \sqrt{147 s^{5} t^{4}} = \sqrt{49 \times 3 \times s^{5} \times t^{4}} \] Next, we can separate the square root: \[ = \sqrt{49} \times \sqrt{3} \times \sqrt{s^{5}} \times \sqrt{t^{4}} \] Calculating each square root: - \( \sqrt{49} = 7 \) - \( \sqrt{3} \) remains as \( \sqrt{3} \). - For \( \sqrt{s^{5}} = \sqrt{s^{4} \cdot s} = \sqrt{s^{4}} \cdot \sqrt{s} = s^{2} \sqrt{s} \) (since \( s^4 \) is a perfect square). - For \( \sqrt{t^{4}} = t^{2} \) (since \( t^4 \) is a perfect square). Putting it all together: \[ \sqrt{147 s^{5} t^{4}} = 7 \times \sqrt{3} \times s^{2} \sqrt{s} \times t^{2} \] Rearranging the terms, we get: \[ = 7 s^{2} t^{2} \sqrt{3s} \] Thus, the simplified form is: \[ \boxed{7 s^{2} t^{2} \sqrt{3s}} \]