Sure, let's simplify the expression [tex]\(\frac{x^{22} \cdot y^{78}}{x^{19} \cdot y^{13}}\)[/tex].
### Step-by-Step Solution:
1. Separate the powers of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in the fraction:
[tex]\[
\frac{x^{22} \cdot y^{78}}{x^{19} \cdot y^{13}} = \frac{x^{22}}{x^{19}} \cdot \frac{y^{78}}{y^{13}}
\][/tex]
2. Simplify the powers of [tex]\(x\)[/tex]:
We have [tex]\(\frac{x^{22}}{x^{19}}\)[/tex].
Using the properties of exponents, where [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
\frac{x^{22}}{x^{19}} = x^{22-19} = x^3
\][/tex]
3. Simplify the powers of [tex]\(y\)[/tex]:
We have [tex]\(\frac{y^{78}}{y^{13}}\)[/tex].
Again, using the properties of exponents:
[tex]\[
\frac{y^{78}}{y^{13}} = y^{78-13} = y^{65}
\][/tex]
4. Combine the simplified terms:
Now, combine the results of the simplified [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms:
[tex]\[
\frac{x^{22} \cdot y^{78}}{x^{19} \cdot y^{13}} = x^3 \cdot y^{65}
\][/tex]
So, the completely simplified expression is:
[tex]\[
\boxed{x^3 y^{65}}
\][/tex]
