To simplify the given expression [tex]\(\frac{z^7 y}{z^5 y^4}\)[/tex], we will follow these steps:
1. Simplify the numerator and the denominator separately:
- In the numerator, we have [tex]\(z^7 y\)[/tex].
- In the denominator, we have [tex]\(z^5 y^4\)[/tex].
2. Combine terms with the same base:
- For the terms with [tex]\(z\)[/tex], we have [tex]\(z^7\)[/tex] in the numerator and [tex]\(z^5\)[/tex] in the denominator.
- For the terms with [tex]\(y\)[/tex], we have [tex]\(y\)[/tex] in the numerator and [tex]\(y^4\)[/tex] in the denominator.
3. Use the properties of exponents to handle the division:
- The property of exponents for division states that [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex].
- Apply this to the terms with [tex]\(z\)[/tex]: [tex]\(\frac{z^7}{z^5} = z^{7-5} = z^2\)[/tex].
- Apply this to the terms with [tex]\(y\)[/tex]: [tex]\(\frac{y}{y^4} = y^{1-4} = y^{-3}\)[/tex].
4. Combine the simplified results:
- The expression now becomes [tex]\(z^2 \cdot y^{-3}\)[/tex].
- Recall that [tex]\(y^{-3}\)[/tex] can be written as [tex]\(\frac{1}{y^3}\)[/tex].
5. Express the final simplified form:
- Therefore, [tex]\(z^2 \cdot \frac{1}{y^3} = \frac{z^2}{y^3}\)[/tex].
So, the simplified form of the given expression is:
[tex]\[
\frac{z^2}{y^3}
\][/tex]
