To simplify the given expression [tex]\(\frac{y^2 z^6}{y^3 z^5}\)[/tex], we will use the properties of exponents. Let's break it down step by step:
1. Separate the expression into parts involving [tex]\(y\)[/tex] and [tex]\(z\)[/tex]:
[tex]\[
\frac{y^2 z^6}{y^3 z^5} = \frac{y^2}{y^3} \cdot \frac{z^6}{z^5}
\][/tex]
2. Simplify the part involving [tex]\(y\)[/tex]:
When dividing expressions with the same base, we subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
\frac{y^2}{y^3} = y^{2-3} = y^{-1}
\][/tex]
3. Simplify the part involving [tex]\(z\)[/tex]:
Similarly, when dividing expressions with the same base, we subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
\frac{z^6}{z^5} = z^{6-5} = z^1 = z
\][/tex]
4. Combine the simplified parts:
Now, we combine the simplified expressions for [tex]\(y\)[/tex] and [tex]\(z\)[/tex]:
[tex]\[
y^{-1} \cdot z
\][/tex]
5. Convert negative exponents to fractions:
A negative exponent indicates the reciprocal of the base with the positive exponent. Therefore,
[tex]\[
y^{-1} = \frac{1}{y}
\][/tex]
6. Write the final simplified expression:
Using the reciprocal property for [tex]\(y^{-1}\)[/tex], we multiply it by [tex]\(z\)[/tex]:
[tex]\[
\frac{1}{y} \cdot z = \frac{z}{y}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\frac{z}{y}
\][/tex]
