select the correct answer.\nsimplify the expression.\n\\(\\sqrt5{224x^{11}y^8}\\)\n\\(\\circ\\) a…

Answer:

C. ( 2x^{2}y\sqrt[5]{7xy^{3}} )

Explanation:

Step1: Factor the radicand

First, factor ( 224x^{11}y^{8} ) into parts that are perfect fifth - powers and the remaining factors. We know that ( 224 = 32\times7=2^{5}\times7 ), ( x^{11}=x^{10 + 1}=(x^{2})^{5}\times x), and ( y^{8}=y^{5+3}=y^{5}\times y^{3} ). So, ( \sqrt[5]{224x^{11}y^{8}}=\sqrt[5]{2^{5}\times7\times(x^{2})^{5}\times x\times y^{5}\times y^{3}} )

Step2: Use the property of radicals ( \sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b} )

According to the property ( \sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b} ) (( a\geq0,b\geq0 ) for even ( n ), ( a,b\in R ) for odd ( n )), we can split the radical: ( \sqrt[5]{2^{5}\times7\times(x^{2})^{5}\times x\times y^{5}\times y^{3}}=\sqrt[5]{2^{5}}\times\sqrt[5]{(x^{2})^{5}}\times\sqrt[5]{y^{5}}\times\sqrt[5]{7xy^{3}} )

Step3: Simplify the perfect fifth - power radicals

Since ( \sqrt[5]{a^{5}} = a ) (for any real number ( a )), we have: ( \sqrt[5]{2^{5}} = 2 ), ( \sqrt[5]{(x^{2})^{5}}=x^{2} ), ( \sqrt[5]{y^{5}} = y ) So, ( \sqrt[5]{2^{5}}\times\sqrt[5]{(x^{2})^{5}}\times\sqrt[5]{y^{5}}\times\sqrt[5]{7xy^{3}}=2\times x^{2}\times y\times\sqrt[5]{7xy^{3}}=2x^{2}[SSE Completed, Client Connection Error][SSE onError error]