To simplify the expression you provided:
\[ \text{index of } 3 \sqrt{y} \left(2y \text{ index of } \sqrt{8y^2} - \text{ index of } 3 \sqrt{y^5} - 4 \text{ index of } 3 \sqrt{8y^2}\right) \]
We need to break this down step by step.
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Understanding the expression:
- The expression is not completely clear because it contains unspecified indices and my notation could be misinterpreted.
- I will interpret them as follows: regarding the "index" of the root, I'll treat "index of \(n \sqrt{x}\)" as referring to the \(n\)-th root, which is \(x^{1/n}\).
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Rewrite the roots as powers:
- \(3 \sqrt{y} = y^{1/3}\)
- \(\sqrt{8y^2} = (8y^2)^{1/2} = 4y\)
- Usual representation of \(3 \sqrt{y^5}\) is \(y^{5/3}\)
- \(3 \sqrt{8y^2} = (8y^2)^{1/3} = 2^{5/3}y^{2/3}\)
Now let's rewrite the entire expression:
\[ y^{1/3} \left(2y \cdot 4y - y^{5/3} - 4 \cdot 2^{5/3} y^{2/3}\right) \]
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Calculate each component:
- \(2y \cdot 4y = 8y^2\)
- Thus, the expression becomes:
\[ y^{1/3} \left(8y^2 - y^{5/3} - 4 \cdot 2^{5/3} y^{2/3}\right) \]
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Distributing \(y^{1/3}\):
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Distributing \(y^{1/3}\) through: \[ y^{1/3} \cdot 8y^2 - y^{1/3} \cdot y^{5/3} - y^{1/3} \cdot 4 \cdot 2^{5/3} y^{2/3} \]
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This becomes:
- \(8y^{2 + 1/3} = 8y^{7/3}\)
- \(- y^{(1/3 + 5/3)} = - y^{6/3} = -y^2\)
- \(-4 \cdot 2^{5/3} y^{(1/3 + 2/3)} = -4 \cdot 2^{5/3} y^{1} = -4 \cdot 2^{5/3} y\)
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Combine everything:
Putting it all together we have:
\[ 8y^{7/3} - y^2 - 4 \cdot 2^{5/3} y \]
- Final expression:
Thus the simplified expression is:
\[ 8y^{7/3} - y^2 - 4 \cdot 2^{5/3} y \]
This is a fully simplified form given the initial interpretation based on your query. If anything needs adjustment or does not align with your intended meaning, please clarify!