Sure, let's simplify the expression [tex]\(\sqrt[4]{768 x^8 y^5}\)[/tex] step-by-step.
Given expression:
[tex]\[
\sqrt[4]{768 x^8 y^5}
\][/tex]
Step 1: Factorize the number inside the radical:
[tex]\[
768 = 256 \times 3 = (4^4) \times 3
\][/tex]
Therefore, we can write:
[tex]\[
768 x^8 y^5 = (4^4 \cdot 3) x^8 y^5
\][/tex]
Step 2: Break the expression under the fourth root into separate fourth roots:
[tex]\[
\sqrt[4]{768 x^8 y^5} = \sqrt[4]{4^4 \cdot 3 x^8 y^5}
\][/tex]
Step 3: Apply the fourth root to each factor separately:
[tex]\[
\sqrt[4]{4^4 \cdot 3 x^8 y^5} = \sqrt[4]{4^4} \cdot \sqrt[4]{3} \cdot \sqrt[4]{x^8} \cdot \sqrt[4]{y^5}
\][/tex]
Step 4: Simplify each fourth root:
[tex]\[
\sqrt[4]{4^4} = 4 \quad \text{(since } (4^4)^{(1/4)} = 4\text{)}
\][/tex]
[tex]\[
\sqrt[4]{x^8} = x^2 \quad \text{(since } (x^8)^{(1/4)} = x^2\text{)}
\][/tex]
[tex]\[
\sqrt[4]{y^5} = y \cdot y^{1/4} \quad \text{(since } y^5 = y \cdot y^4\text{)}
\][/tex]
Now, combining these simplified parts, we get:
[tex]\[
4 \cdot 3^{1/4} \cdot x^2 \cdot y \cdot y^{1/4}
\][/tex]
This can be written as:
[tex]\[
4 \cdot 3^{1/4} \cdot x^2 \cdot y^{1+1/4}
\][/tex]
[tex]\[
= 4 \cdot 3^{1/4} \cdot x^2 \cdot y \cdot y^{1/4}
\][/tex]
Thus, the simplified expression is:
[tex]\[
4 x^2 y \sqrt[4]{3 y}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{4 x^2 y \sqrt[4]{3 y}}
\][/tex]
