To solve the given sum, we will simplify and combine the individual expressions step by step. Here is the problem we need to solve:
[tex]\[ 5 x\left(\sqrt[3]{x^2 y}\right) + 2\left(\sqrt[3]{x^5 y}\right) + 7 x\left(\sqrt[6]{x^2 y}\right) + 7 x^2\left(\sqrt[6]{x y^2}\right) + 7 x^2\left(\sqrt[3]{x y^2}\right) + 7 x\left(\sqrt[3]{x^2 y}\right) \][/tex]
Let’s simplify each term individually:
1. Simplify: [tex]\( 5 x \left( \sqrt[3]{x^2 y} \right) \)[/tex]
[tex]\[
5 x \left( (x^2 y)^{1/3} \right) = 5 x \left( x^{2/3} y^{1/3} \right) = 5 x^{1 + 2/3} y^{1/3} = 5 x^{5/3} y^{1/3}
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2. Simplify: [tex]\( 2 \left( \sqrt[3]{x^5 y} \right) \)[/tex]
[tex]\[
2 \left( (x^5 y)^{1/3} \right) = 2 \left( x^{5/3} y^{1/3} \right)
\][/tex]
3. Simplify: [tex]\( 7 x \left( \sqrt[6]{x^2 y} \right) \)[/tex]
[tex]\[
7 x \left( (x^2 y)^{1/6} \right) = 7 x \left( x^{2/6} y^{1/6} \right) = 7 x \left( x^{1/3} y^{1/6} \right) = 7 x^{4/3} y^{1/6}
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4. Simplify: [tex]\( 7 x^2 \left( \sqrt[6]{x y^2} \right) \)[/tex]
[tex]\[
7 x^2 \left( (x y^2)^{1/6} \right) = 7 x^2 \left( x^{1/6} y^{2/6} \right) = 7 x^2 \left( x^{1/6} y^{1/3} \right) = 7 x^{13/6} y^{1/3}
\][/tex]
5. Simplify: [tex]\( 7 x^2 \left( \sqrt[3]{x y^2} \right) \)[/tex]
[tex]\[
7 x^2 \left( (x y^2)^{1/3} \right) = 7 x^2 \left( x^{1/3} y^{2/3} \right) = 7 x^{7/3} y^{2/3}
\][/tex]
6. Simplify: [tex]\( 7 x \left( \sqrt[3]{x^2 y} \right) \)[/tex]
[tex]\[
7 x \left( (x^2 y)^{1/3} \right) = 7 x \left( x^{2/3} y^{1/3} \right) = 7 x^{5/3} y^{1/3}
\][/tex]
Now we combine the simplified expressions:
[tex]\[
5 x^{5/3} y^{1/3} + 2 x^{5/3} y^{1/3} + 7 x^{4/3} y^{1/6} + 7 x^{13/6} y^{1/3} + 7 x^{7/3} y^{2/3} + 7 x^{5/3} y^{1/3}
\][/tex]
Next step is to group like terms together:
[tex]\[
(5 x^{5/3} y^{1/3} + 2 x^{5/3} y^{1/3} + 7 x^{5/3} y^{1/3}) + 7 x^{4/3} y^{1/6} + 7 x^{13/6} y^{1/3} + 7 x^{7/3} y^{2/3}
\][/tex]
Simplify:
[tex]\[
(5 + 2 + 7)x^{5/3} y^{1/3} + 7 x^{4/3} y^{1/6} + 7 x^{13/6} y^{1/3} + 7 x^{7/3} y^{2/3}
\][/tex]
[tex]\[
14 x^{5/3} y^{1/3} + 7 x^{4/3} y^{1/6} + 7 x^{13/6} y^{1/3} + 7 x^{7/3} y^{2/3}
\][/tex]
So, the sum is:
[tex]\[
7 x^{4/3} y^{1/6} + 7 x^{13/6} y^{1/3} + 7 x^{7/3} y^{2/3} + 14 x^{5/3} y^{1/3} + 2 x^{5/3} y^{1/3}
\][/tex]
