Given:
[tex]\sqrt[3]{875x^5y^9}[/tex]Simplify the expression
[tex]\begin{gathered} \sqrt[3]{875x^5y^9} \\ =\sqrt[3]{7\times125^{}\times x^3\times x^2\times y^3\times y^3\times y^3} \\ =\sqrt[3]{7\times5^3\times x^3\times x^2\times y^3\times y^3\times y^3} \end{gathered}[/tex]Simplify further by taking the cube root of the expression
This gives
[tex]\begin{gathered} \sqrt[3]{7\times5^3\times x^3\times x^2\times y^3\times y^3\times y^3} \\ =5\times x\times y\times y\times y\times\sqrt[3]{7\times x^2} \\ =5xy^3\sqrt[3]{7\times x^2} \\ =5\cdot x\cdot y^3(7^{\frac{1}{3}}\times x^{\frac{2}{3}}) \end{gathered}[/tex]The above result can be simplified as
[tex]\begin{gathered} 5\cdot x\cdot y^3(7^{\frac{1}{3}}\times x^{\frac{2}{3}}) \\ =5^1\cdot7^{\frac{1}{3}}\cdot x^1\cdot x^{\frac{2}{3}}\cdot y^3 \end{gathered}[/tex]Using the same steps, the given expression can be simplified as shown below
[tex]\begin{gathered} \sqrt[3]{875x^5y^9} \\ =\sqrt[3]{7\times125^{}\times x^5\times y^9} \\ =\sqrt[3]{125\times7}\times\sqrt[3]{x^5}\times\sqrt[3]{y^9} \\ =(125\times7)^{\frac{1}{3}}\cdot x^{\frac{5}{3}}\times y^{\frac{9}{3}}^{} \end{gathered}[/tex]Solving the given expression completely
[tex]\begin{gathered} \sqrt[3]{875x^5y^9} \\ =(875x^5y^9)^{\frac{1}{3}} \\ =(125\times7)^{\frac{1}{3}}\times x^{\frac{5}{3}}\times y^{\frac{9}{3}} \\ =125^{\frac{1}{3}}\times7^{\frac{1}{3}}\times x^{(\frac{3}{3}+\frac{2}{3})}\times y^3 \\ =5\times7^{\frac{1}{3}}\times x^1\times x^{\frac{2}{3}}\times y^3 \\ =5xy^3(7^{\frac{1}{3}}\times x^{\frac{2}{3}}) \end{gathered}[/tex]
