Simplify. Assume \( t \) is greater than or equal to zero. \( \sqrt{20 t^{9}} \)

To simplify \( \sqrt{20 t^{9}} \), we can start by breaking it down into its components. The square root of a product can be expressed as the product of the square roots: \[ \sqrt{20 t^{9}} = \sqrt{20} \cdot \sqrt{t^{9}}. \] Next, we simplify \( \sqrt{20} \). The number \( 20 \) can be factored into \( 4 \times 5 \), so: \[ \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5}. \] Now, we simplify \( \sqrt{t^{9}} \). Since we know that \( t^{9} = (t^{4})^{2} \cdot t \), we have: \[ \sqrt{t^{9}} = \sqrt{(t^{4})^{2} \cdot t} = t^{4} \cdot \sqrt{t}. \] Putting it all together, we get: \[ \sqrt{20 t^{9}} = 2\sqrt{5} \cdot t^{4} \cdot \sqrt{t} = 2t^{4}\sqrt{5t}. \] Therefore, the simplified form is: \[ 2t^{4}\sqrt{5t}. \]