To simplify the expression [tex]\(\sqrt{36 y^3}\)[/tex], follow these steps:
1. Expression Under the Square Root:
- We start with [tex]\(\sqrt{36 y^3}\)[/tex].
2. Factorize the Radicand:
- Notice that [tex]\(36 y^3\)[/tex] can be broken down into prime factorization: [tex]\(36 = 6 \times 6 = 6^2\)[/tex].
- So, [tex]\(36 y^3\)[/tex] can be written as [tex]\((6^2) \cdot (y^3)\)[/tex].
3. Separate the Square Root:
- Apply the property of square roots [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex].
- Therefore, [tex]\(\sqrt{36 y^3} = \sqrt{6^2 \cdot y^3} = \sqrt{6^2} \cdot \sqrt{y^3}\)[/tex].
4. Simplify Each Part:
- First, [tex]\(\sqrt{6^2} = 6\)[/tex].
- Second, recall that [tex]\(\sqrt{y^3} = y^{3/2}\)[/tex], which can be split into [tex]\(y^{1} \cdot \sqrt{y} = y \cdot \sqrt{y}\)[/tex].
5. Combine the Results:
- Now, combine these simplifications together: [tex]\(6 \cdot y \cdot \sqrt{y}\)[/tex].
6. Final Simplification:
- The simplified expression is then [tex]\(6 y \sqrt{y}\)[/tex].
Thus, the simplified form of [tex]\(\sqrt{36 y^3}\)[/tex] is [tex]\(6 y \sqrt{y}\)[/tex].
The correct answer from the given choices is none, as the simplified form does not exactly match any of the provided options. But if it had to be formatted matching their ansers it would be something like other than the provided options.
