Sure, let's simplify the expression \(\sqrt[4]{324 x^6 y^8}\) step by step.
1. Prime Factorize the Coefficient:
[tex]\[ 324 = 3^4 \times 4 \][/tex]
2. Break Down the Expression:
[tex]\[ 324 x^6 y^8 = (3^4 \times 4) x^6 y^8 \][/tex]
3. Split the Terms for easier simplification under the fourth root:
[tex]\[ \sqrt[4]{3^4 \times 4 \times x^6 \times y^8} \][/tex]
4. Simplify Each Part Under the Fourth Root:
- [tex]\[ \sqrt[4]{3^4} = 3 \][/tex]
- [tex]\[ \sqrt[4]{x^6} = x^{6/4} = x^{1.5} = x \cdot x^{0.5} = x \sqrt{x} \][/tex]
- [tex]\[ \sqrt[4]{y^8} = y^{8/4} = y^2 \][/tex]
- [tex]\[ \sqrt[4]{4} = \sqrt[4]{2^2} = 2^{1/2} = \sqrt{2} \][/tex]
5. Combine the Results:
[tex]\[ 3 \cdot x \cdot y^2 \cdot \sqrt[4]{4 x^2} \][/tex]
However, note that:
[tex]\[ \sqrt[4]{4 x^2} = \sqrt[4]{2^2 x^2} = (2x)^{1/2} = \sqrt{2x} \][/tex]
6. Simplify the Expression:
Combining the results together, we get:
[tex]\[ 3 x y^2 \sqrt[4]{4 x^2} \][/tex]
Therefore, the simplest form is:
[tex]\(\boxed{3 x y^2 \sqrt[4]{4 x^2}}\)[/tex]
