What is the following sum?

[tex]\[5x \left(\sqrt[3]{x^2 y}\right) + 2 \left(\sqrt[3]{x^5 y}\right)\][/tex]

A. [tex]\(7x \left(\sqrt[6]{x^2 y}\right)\)[/tex]

B. [tex]\(7x^2 \left(\sqrt[6]{x y^2}\right)\)[/tex]

C. [tex]\(7x^2 \left(\sqrt[3]{x y^2}\right)\)[/tex]

D. [tex]\(7x \left(\sqrt[3]{x^2 y}\right)\)[/tex]



Answer :

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]\[ 5 x \left( (x^2 y)^{1/3} \right) = 5 x \left( x^{2/3} y^{1/3} \right) = 5 x^{1 + 2/3} y^{1/3} = 5 x^{5/3} y^{1/3} \][/tex]

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]\[ 7 x \left( (x^2 y)^{1/6} \right) = 7 x \left( x^{2/6} y^{1/6} \right) = 7 x \left( x^{1/3} y^{1/6} \right) = 7 x^{4/3} y^{1/6} \][/tex]

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]