What is the following sum

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What is the following sum? 5x(root(x ^ 2 * y, 3)) + 2(root(x ^ 5 * y, 3)) 7x(root(x ^ 2 * y, 6)) ○ 7x ^ 2 * (root(x * y ^ 2, 6)) 7x ^ 2 * (root(x * y ^ 2, 3)) 7x(root(x ^ 2 * y, 3))

The Correct Answer and Explanation is :

The expression you’re working with involves several terms with roots and exponents, making it a bit complex. Let’s first break it down and simplify the parts step by step. The expression is:

[
5x \cdot \sqrt[3]{x^2y} + 2 \cdot \sqrt[3]{x^5y} \cdot 7x \cdot \sqrt[6]{x^2y} \circ 7x^2 \cdot \sqrt[6]{xy^2} \cdot 7x^2 \cdot \sqrt[3]{xy^2} \cdot 7x \cdot \sqrt[3]{x^2y}
]

First, we identify the root and exponent terms involved:

Step 1: Simplifying the terms with roots and exponents

  1. ( \sqrt[3]{x^2y} ): This is the cube root of ( x^2y ), which can be rewritten as:
    [
    \sqrt[3]{x^2y} = x^{2/3}y^{1/3}
    ]
  2. ( \sqrt[3]{x^5y} ): The cube root of ( x^5y ), which simplifies to:
    [
    \sqrt[3]{x^5y} = x^{5/3}y^{1/3}
    ]
  3. ( \sqrt[6]{x^2y} ): The sixth root of ( x^2y ), which simplifies to:
    [
    \sqrt[6]{x^2y} = x^{2/6}y^{1/6} = x^{1/3}y^{1/6}
    ]
  4. ( \sqrt[6]{xy^2} ): The sixth root of ( xy^2 ), which simplifies to:
    [
    \sqrt[6]{xy^2} = x^{1/6}y^{2/6} = x^{1/6}y^{1/3}
    ]
  5. ( \sqrt[3]{xy^2} ): The cube root of ( xy^2 ), which simplifies to:
    [
    \sqrt[3]{xy^2} = x^{1/3}y^{2/3}
    ]

Step 2: Substituting into the expression

Now, substitute these simplified roots back into the original expression:

[
5x \cdot x^{2/3}y^{1/3} + 2 \cdot x^{5/3}y^{1/3} \cdot 7x \cdot x^{1/3}y^{1/6} \circ 7x^2 \cdot x^{1/6}y^{1/3} \cdot 7x^2 \cdot x^{1/3}y^{2/3} \cdot 7x \cdot x^{2/3}y^{1/3}
]

Step 3: Simplifying terms

Let’s break down the multiplication:

  1. The term ( 5x \cdot x^{2/3}y^{1/3} ) simplifies to:
    [
    5x^{1+2/3}y^{1/3} = 5x^{5/3}y^{1/3}
    ]
  2. The term ( 2 \cdot x^{5/3}y^{1/3} \cdot 7x \cdot x^{1/3}y^{1/6} ) simplifies as follows:
    [
    2 \cdot 7 \cdot x^{5/3+1}y^{1/3+1/6} = 14x^{8/3}y^{1/2}
    ]
  3. The term ( 7x^2 \cdot x^{1/6}y^{1/3} ) simplifies to:
    [
    7x^{2+1/6}y^{1/3} = 7x^{13/6}y^{1/3}
    ]
  4. The term ( 7x^2 \cdot x^{1/3}y^{2/3} ) simplifies to:
    [
    7x^{2+1/3}y^{2/3} = 7x^{7/3}y^{2/3}
    ]
  5. The term ( 7x \cdot x^{2/3}y^{1/3} ) simplifies to:
    [
    7x^{1+2/3}y^{1/3} = 7x^{5/3}y^{1/3}
    ]

Step 4: Combining all terms

Now we combine all the terms together:

[
5x^{5/3}y^{1/3} + 14x^{8/3}y^{1/2} + 7x^{13/6}y^{1/3} + 7x^{7/3}y^{2/3} + 7x^{5/3}y^{1/3}
]

The terms ( 5x^{5/3}y^{1/3} ) and ( 7x^{5/3}y^{1/3} ) combine to:
[
12x^{5/3}y^{1/3}
]

Thus, the final expression is:

[
12x^{5/3}y^{1/3} + 14x^{8/3}y^{1/2} + 7x^{13/6}y^{1/3} + 7x^{7/3}y^{2/3}
]

Conclusion:

This is the simplified version of the given expression. It combines like terms where possible and reduces the complexity of the original equation. The expression is not easily reducible further unless specific values for ( x ) and ( y ) are given.

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