To simplify the given expression step-by-step, we will break it down into its components and then combine like terms.
Given expression:
[tex]\[ 2(\sqrt[4]{18x}) - 2(\sqrt[4]{2y}) + 3(\sqrt[4]{81x}) - 4(\sqrt[4]{32y}) \][/tex]
First, let's simplify each term separately:
1. [tex]\( \sqrt[4]{18x} \)[/tex]
[tex]\[ \sqrt[4]{18x} = \sqrt[4]{18} \cdot \sqrt[4]{x} \][/tex]
Since [tex]\( \sqrt[4]{18} \)[/tex] is a constant factor, we will leave it as is.
2. [tex]\( \sqrt[4]{2y} \)[/tex]
[tex]\[ \sqrt[4]{2y} = \sqrt[4]{2} \cdot \sqrt[4]{y} \][/tex]
Again, [tex]\( \sqrt[4]{2} \)[/tex] is a constant factor.
3. [tex]\( \sqrt[4]{81x} \)[/tex]
[tex]\[ \sqrt[4]{81x} = \sqrt[4]{81} \cdot \sqrt[4]{x} = 3 \cdot \sqrt[4]{x} \][/tex]
We used the property that [tex]\( 81 = 3^4 \)[/tex], so [tex]\( \sqrt[4]{81} = 3 \)[/tex].
4. [tex]\( \sqrt[4]{32y} \)[/tex]
[tex]\[ \sqrt[4]{32y} = \sqrt[4]{32} \cdot \sqrt[4]{y} = 2 \sqrt[4]{2} \cdot \sqrt[4]{y} \][/tex]
We used the property that [tex]\( 32 = 2^5 \)[/tex], therefore [tex]\( \sqrt[4]{32} = \sqrt[4]{2^5} = 2 \sqrt[4]{2} \)[/tex].
Substituting these simplified forms back into the expression, we get:
[tex]\[ 2(\sqrt[4]{18} \cdot \sqrt[4]{x}) - 2(\sqrt[4]{2} \cdot \sqrt[4]{y}) + 3(3 \cdot \sqrt[4]{x}) - 4(2 \sqrt[4]{2} \cdot \sqrt[4]{y}) \][/tex]
[tex]\[ = 2\sqrt[4]{18} \cdot \sqrt[4]{x} - 2\sqrt[4]{2} \cdot \sqrt[4]{y} + 9 \cdot \sqrt[4]{x} - 8\sqrt[4]{2} \cdot \sqrt[4]{y} \][/tex]
Combining like terms:
[tex]\[ = (2\sqrt[4]{18} + 9)\sqrt[4]{x} - (2\sqrt[4]{2} + 8\sqrt[4]{2})\sqrt[4]{y} \][/tex]
[tex]\[ = (2\sqrt[4]{18} + 9)\sqrt[4]{x} - (10\sqrt[4]{2})\sqrt[4]{y} \][/tex]
We need to match this with one of the given choices:
1. [tex]\( 5\sqrt[4]{x} - 4\sqrt[4]{2y} \)[/tex]
2. [tex]\( 5\sqrt[4]{x} - 6\sqrt[4]{2y} \)[/tex]
3. [tex]\( 13\sqrt[4]{x} - 10\sqrt[4]{2y} \)[/tex]
4. [tex]\( 35\sqrt[4]{x} - 18\sqrt[4]{2y} \)[/tex]
Examining the simplified form [tex]\( (2\sqrt[4]{18} + 9)\sqrt[4]{x} - 10\sqrt[4]{2} \cdot \sqrt[4]{y} \)[/tex] against the choices given, none seem to precisely match it. Therefore, the correct conclusion in reducing this matched expression would be that none of the choices align perfectly with the simplification process we discussed. So, none of the given options correctly match the simplified expression.
