Let's simplify the expression step-by-step:
[tex]\[
2(\sqrt[4]{16 x})-2(\sqrt[4]{2 y})+3(\sqrt[4]{81 x})-4(\sqrt[4]{32 y})
\][/tex]
1. Simplify each term separately:
- For the first term:
[tex]\[
2(\sqrt[4]{16 x}) = 2((16 x)^{1/4})
\][/tex]
Notice that [tex]\(16 = 2^4\)[/tex]:
[tex]\[
2((2^4 x)^{1/4}) = 2(2 \cdot x^{1/4}) = 2 \cdot 2 \cdot x^{1/4} = 4 x^{1/4}
\][/tex]
- For the second term:
[tex]\[
-2(\sqrt[4]{2 y}) = -2((2 y)^{1/4})
\][/tex]
This remains:
[tex]\[
-2 \cdot (2 y)^{1/4}
\][/tex]
- For the third term:
[tex]\[
3(\sqrt[4]{81 x}) = 3((81 x)^{1/4})
\][/tex]
Notice that [tex]\(81 = 3^4\)[/tex]:
[tex]\[
3((3^4 x)^{1/4}) = 3(3 \cdot x^{1/4}) = 3 \cdot 3 \cdot x^{1/4} = 9 x^{1/4}
\][/tex]
- For the fourth term:
[tex]\[
-4(\sqrt[4]{32 y}) = -4((32 y)^{1/4})
\][/tex]
Notice that [tex]\(32 = 2^5\)[/tex]:
[tex]\[
-4((2^5 y)^{1/4}) = -4(2^{5/4} y^{1/4})
\][/tex]
Simplify [tex]\(2^{5/4}\)[/tex]:
[tex]\[
2^{5/4} = 2 \cdot 2^{1/4} = 2 \cdot (2 y)^{1/4}
\][/tex]
Thus,
[tex]\[
-4 \cdot 2 \cdot (2 y)^{1/4} = -8 (2 y)^{1/4}
\][/tex]
2. Combine all simplified terms:
[tex]\[
4 x^{1/4} - 2 (2 y)^{1/4} + 9 x^{1/4} - 8 (2 y)^{1/4}
\][/tex]
3. Combine like terms [tex]\(x^{1/4}\)[/tex] and [tex]\((2 y)^{1/4}\)[/tex]:
[tex]\[
4 x^{1/4} + 9 x^{1/4} = 13 x^{1/4}
\][/tex]
[tex]\[
-2 (2 y)^{1/4} - 8 (2 y)^{1/4} = -10 (2 y)^{1/4}
\][/tex]
4. Final simplified expression:
[tex]\[
13 x^{1/4} - 10 (2 y)^{1/4}
\][/tex]
Therefore, the simplified form of the expression is:
[tex]\[
\boxed{13 (\sqrt[4]{x}) - 10 (\sqrt[4]{2 y})}
\][/tex]
