Questions: Convert the given product to a single radical with lowest index. Assume that all variables represent positive numbers. √[16]11 m^3 n · √[4]3 m^5 n^3 √[16]11 m^3 n · √[4]3 m^5 n^3= (Simplify your answer. Type an exact answer, using radicals as needed.)

Solution Steps

We start by rewriting the given radicals in exponential form: \[ \sqrt[16]{11 m^{3} n} = 11^{\frac{1}{16}} m^{\frac{3}{16}} n^{\frac{1}{16}} \] \[ \sqrt[4]{3 m^{5} n^{3}} = 3^{\frac{1}{4}} m^{\frac{5}{4}} n^{\frac{3}{4}} \]

The indices of the radicals are \(16\) and \(4\). The least common multiple (LCM) of \(16\) and \(4\) is \(16\).

We express both terms with the common index of \(16\): \[ \sqrt[16]{11 m^{3} n} = 11^{\frac{1}{16}} m^{\frac{3}{16}} n^{\frac{1}{16}} \] \[ \sqrt[16]{3 m^{5} n^{3}} = 3^{\frac{1}{4}} m^{\frac{5}{4}} n^{\frac{3}{4}} = 3^{\frac{4}{16}} m^{\frac{20}{16}} n^{\frac{12}{16}} \]

Now we multiply the two expressions: \[ 11^{\frac{1}{16}} \cdot 3^{\frac{4}{16}} \cdot m^{\frac{3}{16}} \cdot m^{\frac{20}{16}} \cdot n^{\frac{1}{16}} \cdot n^{\frac{12}{16}} = 11^{\frac{1}{16}} \cdot 3^{\frac{4}{16}} \cdot m^{\frac{23}{16}} \cdot n^{\frac{13}{16}} \]

The product simplifies to: \[ 11^{\frac{1}{16}} \cdot 3^{\frac{4}{16}} \cdot m^{\frac{23}{16}} \cdot n^{\frac{13}{16}} \]

Finally, we express the simplified product in radical form: \[ \sqrt[16]{11} \cdot \sqrt[16]{3^{4}} \cdot m^{\frac{23}{16}} \cdot n^{\frac{13}{16}} = \sqrt[16]{11} \cdot 3^{\frac{1}{4}} \cdot m^{\frac{23}{16}} \cdot n^{\frac{13}{16}} \]

Final Answer