What is the following product? Assume x ≥ 0\sqrt[3]{x^{2} } . \sqrt[4]{x^{3} }A. x\sqrt{x}B. \sqrt[12]{x^{5} }C. x(\sqrt[12]{x^{5} } )D. x6

For this case we must multiply the following expression:[tex]\sqrt [3] {x ^ 2} * \sqrt [4] {x ^ 3}[/tex]By definition of properties of otencias and roots we have:[tex]\sqrt [n] {a ^ m} = a ^ {\frac {m} {n}}[/tex]We rewrite the terms of the expression:[tex]\sqrt [3] {x ^ 2} = (x ^ 2) ^ {\frac {1} {3}} = (x ^ 2) ^ {\frac {4} {12}}\\\sqrt [4] {x ^ 3} = (x ^ 3) ^ {\frac {1} {4}} = (x ^ 3) ^ {\frac {3} {12}}[/tex]So, we have:[tex](x ^ 2) ^ {\frac {4} {12}} * (x ^ 3) ^ {\frac {3} {12}} =[/tex]Applying the above definition we have:[tex]\sqrt [12] {(x ^ 2) ^ 4} * \sqrt [12] {(x ^ 3) ^ 3} =[/tex]We multiply the exponents:[tex]\sqrt [12] {x ^ 8} * \sqrt [12] {x ^ 9} =[/tex]We combine using the product rule for radicals.[tex]\sqrt [12] {x ^ 8 * x ^ 9} =[/tex]By definition of multiplication properties of powers of the same base, we put the same base and add the exponents:[tex]\sqrt [12] {x ^ {8 + 9}} =\\\sqrt [12] {x ^ {17}} =\\\sqrt [12] {x ^ {12} * x ^ 5} =\\x \sqrt [12] {x ^ 5}[/tex]Answer:Option C