Answer:
A
Step-by-step explanation:
Given expression:
[tex]\sqrt[4]{768x^8y^5}[/tex]
Using rule [tex]\sqrt{ab} =\sqrt{a} \cdot \sqrt{b}[/tex]
[tex]=\sqrt[4]{768}\cdot \sqrt[4]{x^8} \cdot\sqrt[4]{y^5}[/tex]
Prime factorization of 768:
[tex]=\sqrt[4]{2^8\cdot3} \cdot \sqrt[4]{x^8} \cdot\sqrt[4]{y^5}[/tex]
[tex]=\sqrt[4]{2^8} \cdot \sqrt[4]{3} \cdot \sqrt[4]{x^8} \cdot\sqrt[4]{y^5}[/tex]
Using rule [tex]a^{b+c}=a^ba^c[/tex] :
[tex]=\sqrt[4]{2^8} \cdot \sqrt[4]{3} \cdot \sqrt[4]{x^8} \cdot\sqrt[4]{y^{4+1}}[/tex]
[tex]=\sqrt[4]{2^8} \cdot \sqrt[4]{3} \cdot \sqrt[4]{x^8} \cdot\sqrt[4]{y^4y^1}}[/tex]
[tex]=\sqrt[4]{2^8} \cdot \sqrt[4]{3} \cdot \sqrt[4]{x^8} \cdot\sqrt[4]{y^4} \cdot\sqrt[4]{y}[/tex]
Using rule [tex]\sqrt[a]{b^c} =b^{\frac{c}{a}}[/tex] :
[tex]=2^{\frac84}} \cdot \sqrt[4]{3} \cdot x^{\frac84}\cdot y^{\frac44} \cdot \sqrt[4]{y}[/tex]
[tex]=2^2 \cdot \sqrt[4]{3} \cdot x^2\cdot y \cdot \sqrt[4]{y}[/tex]
[tex]=4 \cdot \sqrt[4]{3} \cdot x^2\cdot y \cdot \sqrt[4]{y}[/tex]
Gather like terms:
[tex]=4 \cdot x^2\cdot y\cdot \sqrt[4]{3} \cdot \sqrt[4]{y}[/tex]
[tex]=4x^2y \sqrt[4]{3y}[/tex]
