Sure, let's break down the given expression step by step and simplify it:
The expression we need to simplify is:
[tex]\[
\frac{3^{3x+2} - 3^{3x+1}}{27^x \times 6}
\][/tex]
Step 1: Rewrite the bases in terms of \(3\).
First, note that \(27\) is a power of \(3\):
[tex]\[
27 = 3^3
\][/tex]
Therefore, \(27^x\) can be written as \((3^3)^x = 3^{3x}\).
Step 2: Substitute and simplify the denominator.
Replace \(27^x\) with \(3^{3x}\) in the denominator:
[tex]\[
27^x \times 6 = 3^{3x} \times 6
\][/tex]
Step 3: Simplify the numerator.
The numerator is \(3^{3x+2} - 3^{3x+1}\). We can factor out the common term \(3^{3x+1}\) from both terms in the numerator:
[tex]\[
3^{3x+2} - 3^{3x+1} = 3^{3x+1}(3 - 1)
\][/tex]
Simplify the expression within the parentheses:
[tex]\[
(3-1) = 2
\][/tex]
Thus the numerator simplifies to:
[tex]\[
3^{3x+1} \times 2
\][/tex]
Step 4: Combine the simplified numerator and denominator.
Rewrite the original expression with the simplified numerator and denominator:
[tex]\[
\frac{3^{3x+1} \times 2}{3^{3x} \times 6}
\][/tex]
Step 5: Simplify the fraction.
First, we can cancel the common term \(3^{3x}\) in the numerator and denominator:
[tex]\[
\frac{3^{3x+1} \times 2}{3^{3x} \times 6} = \frac{3 \times 2}{6}
\][/tex]
Next, compute the remaining fraction:
[tex]\[
\frac{3 \times 2}{6} = \frac{6}{6} = 1
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\boxed{1}
\][/tex]
