To solve the given equation
[tex]\[
\frac{9^n \cdot 3^2 \cdot 3^n - 27^n}{(3^m \cdot 2)^{27}} = 3^{-3}
\][/tex]
let's start by simplifying both the numerator and the denominator.
First, simplify the expressions in the numerator:
1. [tex]\(9^n\)[/tex] can be written as [tex]\((3^2)^n = 3^{2n}\)[/tex].
2. [tex]\(3^2\)[/tex] is [tex]\(9\)[/tex] or [tex]\(3^2\)[/tex].
3. [tex]\(3^n\)[/tex] is itself [tex]\(3^n\)[/tex].
4. [tex]\(27^n\)[/tex] can be written as [tex]\((3^3)^n = 3^{3n}\)[/tex].
Using these equivalences, the numerator becomes:
[tex]\[
9^n \cdot 3^2 \cdot 3^n - 27^n = 3^{2n} \cdot 3^2 \cdot 3^n - 3^{3n}
\][/tex]
Next, combine the exponents of 3 in the simplified numerator:
[tex]\[
3^{2n} \cdot 3^2 \cdot 3^n = 3^{(2n + 2 + n)} = 3^{(3n + 2)}
\][/tex]
Thus, the numerator simplifies to:
[tex]\[
3^{3n + 2} - 3^{3n}
\][/tex]
Now, simplify the denominator:
[tex]\[
(3^m \cdot 2)^{27}
\][/tex]
Raise each term to the power of 27:
[tex]\[
3^{27m} \cdot 2^{27}
\][/tex]
Now the equation can be rewritten as:
[tex]\[
\frac{3^{3n + 2} - 3^{3n}}{3^{27m} \cdot 2^{27}} = 3^{-3}
\][/tex]
For the right-hand side to equal [tex]\(3^{-3}\)[/tex], the terms involving powers of 3 in the numerator and denominator must be consistent in their exponents.
To equate the exponents of 3, let's express the powers clearly.
From the left-hand side, focus on the most significant terms involving powers of 3. Collect the exponent terms:
[tex]\[
3^{3n + 2} - 3^{3n}
\][/tex]
If we factor out [tex]\(3^{3n}\)[/tex] from the numerator:
[tex]\[
3^{3n}(3^2 - 1) = 3^{3n} \cdot 8 = 3^{3n} \cdot 2^3
\][/tex]
Thus, the equation becomes:
[tex]\[
\frac{3^{3n} \cdot 2^3}{3^{27m} \cdot 2^{27}} = 3^{-3}
\][/tex]
Now, simplifying the fractions:
[tex]\[
\frac{3^{3n} \cdot 2^3}{3^{27m} \cdot 2^{27}} = \frac{3^{3n}}{3^{27m}} \cdot \frac{2^3}{2^{27}} = 3^{3n-27m} \cdot 2^{3-27}
\][/tex]
We know this equals [tex]\(3^{-3}\)[/tex]. So that gives us two separate equations for the exponents. The exponent of 3:
[tex]\[
3n - 27m = -3
\][/tex]
The exponent of 2:
[tex]\[
3 - 27 = -24
\][/tex]
These two should hold true. For the exponent of 3:
[tex]\[
3n - 27m = -3 \implies n - 9m = -1 \implies m - n = 1
\][/tex]
Therefore, we have shown that:
[tex]\[
m - n = 1
\][/tex]
