Question

simplify: index of 3 √y(2y index of √8y^2- index of 3 √y^5-4 index of 3√8 y^2

Answer 1

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.

1. **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 \(x^{1/n}\).

2. **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:

\[
y^{1/3} \left(2y \cdot 4y - y^{5/3} - 4 \cdot 2^{5/3} y^{2/3}\right)
\]

3. **Calculate each component**:

- \(2y \cdot 4y = 8y^2\)
- Thus, the expression becomes:

\[
y^{1/3} \left(8y^2 - y^{5/3} - 4 \cdot 2^{5/3} y^{2/3}\right)
\]

4. **Distributing \(y^{1/3}\)**:

- Distributing \(y^{1/3}\) through:
\[
y^{1/3} \cdot 8y^2 - y^{1/3} \cdot y^{5/3} - y^{1/3} \cdot 4 \cdot 2^{5/3} y^{2/3}
\]

- This becomes:
- \(8y^{2 + 1/3} = 8y^{7/3}\)
- \(- y^{(1/3 + 5/3)} = - y^{6/3} = -y^2\)
- \(-4 \cdot 2^{5/3} y^{(1/3 + 2/3)} = -4 \cdot 2^{5/3} y^{1} = -4 \cdot 2^{5/3} y\)

5. **Combine everything**:

Putting it all together we have:

\[
8y^{7/3} - y^2 - 4 \cdot 2^{5/3} y
\]

6. **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!