Answer :
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]
[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]
