The sum can be evaluated using the rules of indices to simplify the terms.
- The sum [tex]\mathbf{\sqrt[3]{125 \cdot x^{10}\cdot y^{13}} + \sqrt[3]{27 \cdot x^{10} \cdot y^{13}}}[/tex] = [tex]\underline{8\cdot x^{3 } \cdot y^{4 } \cdot \sqrt[3]{x\cdot y}}[/tex]
Reasons:
The given sum is presented as follows;
[tex]\mathbf{\sqrt[3]{125 \cdot x^{10}\cdot y^{13}} + \sqrt[3]{27 \cdot x^{10} \cdot y^{13}}}[/tex]
Simplifying gives;
[tex]\sqrt[3]{5^3\cdot x^{3 \times 3} \cdot x\cdot y^{3 \times 4}\cdot y} + \sqrt[3]{3^3\cdot x^{3 \times 3} \cdot x\cdot y^{3 \times 4}\cdot y}[/tex]
[tex]\mathbf{\sqrt[3]{(5\cdot x^{3 } \cdot y^{4 })^3 \cdot x\cdot y} + \sqrt[3]{(3\cdot x^{3 } \cdot y^{4 })^3 \cdot x\cdot y}}[/tex]
- [tex](5\cdot x^{3 } \cdot y^{4 }) \cdot \sqrt[3]{x\cdot y} + (3\cdot x^{3 } \cdot y^{4 })\cdot \sqrt[3]{x\cdot y}[/tex]
Factorizing gives;
- [tex](5\cdot x^{3 } \cdot y^{4 }) \cdot \sqrt[3]{x\cdot y} + (3\cdot x^{3 } \cdot y^{4 })\cdot \sqrt[3]{x\cdot y} = \sqrt[3]{x\cdot y} \cdot \left((5\cdot x^{3 } \cdot y^{4 }) \cdot + (3\cdot x^{3 } \cdot y^{4 }) \right)[/tex]
- [tex]\sqrt[3]{x\cdot y} \cdot \left((5\cdot x^{3 } \cdot y^{4 }) \cdot + (3\cdot x^{3 } \cdot y^{4 }) \right) = \mathbf{ \sqrt[3]{x\cdot y} \cdot 8\cdot x^{3 } \cdot y^{4 } = 8\cdot x^{3 } \cdot y^{4 } \cdot \sqrt[3]{x\cdot y}}[/tex]
Which gives;
- [tex]\underline{\mathbf{\sqrt[3]{125 \cdot x^{10}\cdot y^{13}} + \sqrt[3]{27 \cdot x^{10} \cdot y^{13}}} = 8\cdot x^{3 } \cdot y^{4 } \cdot \sqrt[3]{x\cdot y}}[/tex]
The correct option is therefore;
[tex]\underline{ \mathbf{8\cdot x^{3 } \cdot y^{4 } \cdot \sqrt[3]{x\cdot y}}}[/tex] which can be expressed as follows;
- 8 x cubed y superscript 4 Baseline(RootIndex 3 StartRoot x y EndRoot)
Learn more about rules of indices here:
https://brainly.com/question/10290723
