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Answer :
To solve the given expression:
[tex]\[ 2\left(\sqrt[3]{16 x^3 y}\right) + 4\left(\sqrt[3]{54 x^6 y^5}\right) + 4 x(\sqrt[3]{2 y}) + 12 x^2 y\left(\sqrt[3]{2 y^2}\right) + 8 x(\sqrt[3]{x y}) + 12 x^3 y^2(\sqrt[3]{6 y}) + 16 x^3 y\left(\sqrt[3]{2 y^2}\right) + 48 x^3 y(\sqrt[3]{2 y}) \][/tex]
we will break it down into individual components and simplify each:
1. Simplify each cube root:
- For [tex]\(\sqrt[3]{16 x^3 y}\)[/tex]:
The expression inside the cube root is [tex]\(16 x^3 y = (2^4 x^3 y)\)[/tex].
The cube root becomes: [tex]\(2^{4/3}x(y)^{1/3}\)[/tex].
- For [tex]\(\sqrt[3]{54 x^6 y^5}\)[/tex]:
The expression inside the cube root is [tex]\(54 x^6 y^5\)[/tex].
The cube root becomes: [tex]\((2 \times 3^3)^{1/3} x^2 y^{5/3}\)[/tex].
- For [tex]\(\sqrt[3]{2 y}\)[/tex]:
The cube root is: [tex]\(2^{1/3} y^{1/3}\)[/tex].
- For [tex]\(\sqrt[3]{2 y^2}\)[/tex]:
The cube root is: [tex]\(2^{1/3} y^{2/3}\)[/tex].
- For [tex]\(\sqrt[3]{x y}\)[/tex]:
The cube root is: [tex]\(x^{1/3} y^{1/3}\)[/tex].
- For [tex]\(\sqrt[3]{6 y}\)[/tex]:
The cube root is: [tex]\((2 \times 3)^{1/3} y^{1/3}\)[/tex].
2. Multiply each simplified cube root by its coefficient:
- [tex]\(2 \times 2^{4/3}(x^3 y)^{1/3} = 4 \times 2^{1/3} \times (x^3 y)^{1/3}\)[/tex].
- [tex]\(4 \times 12 \times 2^{1/3} x^2 y^{5/3}\)[/tex].
- [tex]\(4 x \times 2^{1/3} y^{1/3}\)[/tex].
- [tex]\(12 x^2 y \times 2^{1/3} y^{2/3}\)[/tex].
- [tex]\(8 x \times x^{1/3} y^{1/3}\)[/tex].
- [tex]\(12 x^3 y^2 \times 6^{1/3} y^{1/3}\)[/tex].
- [tex]\(16 x^3 y \times 2^{1/3} y^{2/3}\)[/tex].
- [tex]\(48 x^3 y \times 2^{1/3} y^{1/3}\)[/tex].
3. Add all parts together to get the final sum*:
[tex]\[
12 \times 6^{1/3} x^3 y^{7/3} + 48 \times 2^{1/3} x^3 y^{4/3} + 16 \times 2^{1/3} x^3 y y^{2/3} + 12 \times 2^{1/3} x^2 y y^{2/3} + 4 \times 2^{1/3} x y^{1/3} + 8 x (x y)^{1/3} + 4 \times 2^{1/3} (x^3 y)^{1/3} + 12 \times 2^{1/3} (x^6 y^5)^{1/3}
\][/tex]
4. Combine all similar terms:
The expression combines through its like terms to form a more complex polynomial in terms of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and their roots. The exact algebraic simplification depends on common factors and the chosen format for the solution.
This will provide the complete simplified sum of the initial expression.
[tex]\[ 2\left(\sqrt[3]{16 x^3 y}\right) + 4\left(\sqrt[3]{54 x^6 y^5}\right) + 4 x(\sqrt[3]{2 y}) + 12 x^2 y\left(\sqrt[3]{2 y^2}\right) + 8 x(\sqrt[3]{x y}) + 12 x^3 y^2(\sqrt[3]{6 y}) + 16 x^3 y\left(\sqrt[3]{2 y^2}\right) + 48 x^3 y(\sqrt[3]{2 y}) \][/tex]
we will break it down into individual components and simplify each:
1. Simplify each cube root:
- For [tex]\(\sqrt[3]{16 x^3 y}\)[/tex]:
The expression inside the cube root is [tex]\(16 x^3 y = (2^4 x^3 y)\)[/tex].
The cube root becomes: [tex]\(2^{4/3}x(y)^{1/3}\)[/tex].
- For [tex]\(\sqrt[3]{54 x^6 y^5}\)[/tex]:
The expression inside the cube root is [tex]\(54 x^6 y^5\)[/tex].
The cube root becomes: [tex]\((2 \times 3^3)^{1/3} x^2 y^{5/3}\)[/tex].
- For [tex]\(\sqrt[3]{2 y}\)[/tex]:
The cube root is: [tex]\(2^{1/3} y^{1/3}\)[/tex].
- For [tex]\(\sqrt[3]{2 y^2}\)[/tex]:
The cube root is: [tex]\(2^{1/3} y^{2/3}\)[/tex].
- For [tex]\(\sqrt[3]{x y}\)[/tex]:
The cube root is: [tex]\(x^{1/3} y^{1/3}\)[/tex].
- For [tex]\(\sqrt[3]{6 y}\)[/tex]:
The cube root is: [tex]\((2 \times 3)^{1/3} y^{1/3}\)[/tex].
2. Multiply each simplified cube root by its coefficient:
- [tex]\(2 \times 2^{4/3}(x^3 y)^{1/3} = 4 \times 2^{1/3} \times (x^3 y)^{1/3}\)[/tex].
- [tex]\(4 \times 12 \times 2^{1/3} x^2 y^{5/3}\)[/tex].
- [tex]\(4 x \times 2^{1/3} y^{1/3}\)[/tex].
- [tex]\(12 x^2 y \times 2^{1/3} y^{2/3}\)[/tex].
- [tex]\(8 x \times x^{1/3} y^{1/3}\)[/tex].
- [tex]\(12 x^3 y^2 \times 6^{1/3} y^{1/3}\)[/tex].
- [tex]\(16 x^3 y \times 2^{1/3} y^{2/3}\)[/tex].
- [tex]\(48 x^3 y \times 2^{1/3} y^{1/3}\)[/tex].
3. Add all parts together to get the final sum*:
[tex]\[
12 \times 6^{1/3} x^3 y^{7/3} + 48 \times 2^{1/3} x^3 y^{4/3} + 16 \times 2^{1/3} x^3 y y^{2/3} + 12 \times 2^{1/3} x^2 y y^{2/3} + 4 \times 2^{1/3} x y^{1/3} + 8 x (x y)^{1/3} + 4 \times 2^{1/3} (x^3 y)^{1/3} + 12 \times 2^{1/3} (x^6 y^5)^{1/3}
\][/tex]
4. Combine all similar terms:
The expression combines through its like terms to form a more complex polynomial in terms of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and their roots. The exact algebraic simplification depends on common factors and the chosen format for the solution.
This will provide the complete simplified sum of the initial expression.
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