6. Rewrite each radical by extracting all possible roots, and write the final answer in radical form. \( \begin{array}{ll}\text { a. } \sqrt{16 x^{6}} & \text { b. }-\sqrt{8 v^{3}} \\ \text { c. } \sqrt{d^{3} f^{4}} & \text { d. } \sqrt{h^{4} j^{6}} \\ \text { e. } \sqrt{25 a^{2} b^{8} c^{10}} & \text { f. } \sqrt[4]{81 x^{5} y^{12}} \\ \text { g. } \sqrt[3]{(x+3)^{9}} & \text { h. } \sqrt{(x+3)^{2}}\end{array} \)

To rewrite each radical by extracting all possible roots, we will simplify each expression step by step. ### a. \( \sqrt{16 x^{6}} \) 1. The square root of \(16\) is \(4\). 2. The square root of \(x^{6}\) is \(x^{3}\) (since \(\sqrt{x^{2n}} = x^{n}\)). 3. Therefore, \( \sqrt{16 x^{6}} = 4x^{3} \). ### b. \( -\sqrt{8 v^{3}} \) 1. The square root of \(8\) can be simplified as \( \sqrt{4 \cdot 2} = 2\sqrt{2} \). 2. The square root of \(v^{3}\) is \(v^{1} \sqrt{v} = v\sqrt{v}\). 3. Therefore, \( -\sqrt{8 v^{3}} = -2v\sqrt{2v} \). ### c. \( \sqrt{d^{3} f^{4}} \) 1. The square root of \(d^{3}\) is \(d^{1} \sqrt{d} = d\sqrt{d}\). 2. The square root of \(f^{4}\) is \(f^{2}\). 3. Therefore, \( \sqrt{d^{3} f^{4}} = df^{2}\sqrt{d} \). ### d. \( \sqrt{h^{4} j^{6}} \) 1. The square root of \(h^{4}\) is \(h^{2}\). 2. The square root of \(j^{6}\) is \(j^{3}\). 3. Therefore, \( \sqrt{h^{4} j^{6}} = h^{2} j^{3} \). ### e. \( \sqrt{25 a^{2} b^{8} c^{10}} \) 1. The square root of \(25\) is \(5\). 2. The square root of \(a^{2}\) is \(a\). 3. The square root of \(b^{8}\) is \(b^{4}\). 4. The square root of \(c^{10}\) is \(c^{5}\). 5. Therefore, \( \sqrt{25 a^{2} b^{8} c^{10}} = 5abc^{5} \). ### f. \( \sqrt[4]{81 x^{5} y^{12}} \) 1. The fourth root of \(81\) is \(3\). 2. The fourth root of \(x^{5}\) is \(x^{1} \sqrt[4]{x} = x\sqrt[4]{x}\). 3. The fourth root of \(y^{12}\) is \(y^{3}\). 4. Therefore, \( \sqrt[4]{81 x^{5} y^{12}} = 3xy^{3}\sqrt[4]{x} \). ### g. \( \sqrt[3]{(x+3)^{9}} \) 1. The cube root of \((x+3)^{9}\) is \((x+3)^{3}\). 2. Therefore, \( \sqrt[3]{(x+3)^{9}} = (x+3)^{3} \). ### h. \( \sqrt{(x+3)^{2}} \) 1. The square root of \((x+3)^{2}\) is \((x+3)\). 2. Therefore, \( \sqrt{(x+3)^{2}} = (x+3) \). ### Final Answers - a. \( 4x^{3} \) - b. \( -2v\sqrt{2v} \) - c. \( df^{2}\sqrt{d} \) - d. \( h^{2} j^{3} \) - e. \( 5abc^{5} \) - f. \( 3xy^{3}\sqrt[4]{x} \) - g. \( (x+3)^{3} \) - h. \( (x+3) \)