Sure, let's take a detailed step-by-step approach to simplify the given expression:
[tex]\[
\frac{6^n \cdot 6^{4 n}}{6^{5 n} \cdot 6^2}
\][/tex]
### Step 1: Apply the properties of exponents
First, let's recall the properties of exponents:
1. [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]
2. [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]
### Step 2: Simplify the numerator
Combine the exponents in the numerator:
[tex]\[
6^n \cdot 6^{4 n} = 6^{n + 4n} = 6^{5n}
\][/tex]
### Step 3: Simplify the denominator
Combine the exponents in the denominator:
[tex]\[
6^{5 n} \cdot 6^2 = 6^{5n + 2}
\][/tex]
### Step 4: Combine the simplified numerator and denominator
Now, you have:
[tex]\[
\frac{6^{5n}}{6^{5n + 2}}
\][/tex]
### Step 5: Apply the properties of exponents to the fraction
Using the second property ([tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]), we subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
6^{5n} / 6^{5n + 2} = 6^{5n - (5n + 2)} = 6^{5n - 5n - 2} = 6^{-2}
\][/tex]
### Step 6: Simplify the exponent
So we have:
[tex]\[
6^{-2}
\][/tex]
### Conclusion
The simplified form of the given expression is:
[tex]\[
6^{-2}
\][/tex]
Or, if you prefer, you can also express it as:
[tex]\[
\frac{1}{6^2} = \frac{1}{36}
\][/tex]
So, the final answer is [tex]\(6^{-2}\)[/tex] or [tex]\(\frac{1}{36}\)[/tex].
