To simplify the given expression [tex]\(\frac{y^{-2}}{6 y^3}\)[/tex], let's go through the problem step-by-step:
1. Simplify the Exponents:
The numerator of the fraction has [tex]\(y^{-2}\)[/tex], and the denominator has [tex]\(6 y^3\)[/tex]. We need to combine the exponents of [tex]\(y\)[/tex]
[tex]\[
y^{-2} \text{ and } y^3.
\][/tex]
When we divide with the same base, we subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
y^{-2 - 3} = y^{-5}.
\][/tex]
2. Combine with the Coefficient:
The coefficient 6 remains in the denominator. Thus, the expression becomes:
[tex]\[
\frac{y^{-5}}{6}.
\][/tex]
This can be rewritten as:
[tex]\[
\frac{1}{6} y^{-5}.
\][/tex]
3. Express in Standard Form:
To make the expression cleaner and in standard mathematical form, we typically write it as:
[tex]\[
\frac{1}{6 y^5}.
\][/tex]
4. Cross-check with the Choices:
Let's compare our simplified result with the given choices:
- [tex]\(\frac{y^7}{6}\)[/tex]
- [tex]\(\frac{1}{6 y^7}\)[/tex]
- [tex]\(\frac{1}{6 y^3}\)[/tex]
- [tex]\(-\frac{1}{6 y^7}\)[/tex]
The simplified form [tex]\(\frac{1}{6 y^5}\)[/tex] doesn't match any of the provided options exactly. Therefore, none of the provided choices is the correct answer.
So, the correct simplified form of [tex]\(\frac{y^{-2}}{6 y^3}\)[/tex] is:
[tex]\[
\frac{1}{6 y^5}.
\][/tex]
