Certainly! Let's simplify the given expression step-by-step:
Given expression:
[tex]\[
\frac{x^6 z^2}{x^6 z^3}
\][/tex]
1. Cancel identical terms in the numerator and denominator:
The term [tex]\(x^6\)[/tex] appears both in the numerator and the denominator. Since any nonzero number divided by itself is 1, we can cancel out [tex]\(x^6\)[/tex] from both the numerator and the denominator:
[tex]\[
\frac{x^6 z^2}{x^6 z^3} = \frac{z^2}{z^3}
\][/tex]
2. Simplify the remaining fraction [tex]\(\frac{z^2}{z^3}\)[/tex]:
When dividing expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
\frac{z^2}{z^3} = z^{2-3} = z^{-1}
\][/tex]
3. Rewrite the expression [tex]\(z^{-1}\)[/tex] as a fraction:
Recall that [tex]\(z^{-1}\)[/tex] is equivalent to [tex]\(\frac{1}{z}\)[/tex]:
[tex]\[
z^{-1} = \frac{1}{z}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\frac{1}{z}
\][/tex]
