To solve the equation [tex]\(7 + 42 \cdot 3^{2 - 3a} = 14 \cdot 3^{-2a} + 7\)[/tex], follow these steps:
1. Simplify the Equation:
[tex]\[
7 + 42 \cdot 3^{2 - 3a} = 14 \cdot 3^{-2a} + 7
\][/tex]
We see that both sides of the equation have a [tex]\(7\)[/tex]. Subtract [tex]\(7\)[/tex] from both sides to simplify:
[tex]\[
42 \cdot 3^{2 - 3a} = 14 \cdot 3^{-2a}
\][/tex]
2. Isolate the exponentials:
Divide both sides by [tex]\(14\)[/tex] to further simplify:
[tex]\[
3 \cdot 3^{2 - 3a} = 3^{-2a}
\][/tex]
Simplify the left side:
[tex]\[
3^{1 + (2 - 3a)} = 3^{-2a}
\][/tex]
Combine the exponents since the bases are the same (both are powers of 3):
[tex]\[
3^{3 - 3a} = 3^{-2a}
\][/tex]
3. Set the exponents equal to each other:
Since the bases are the same, the exponents must be equal:
[tex]\[
3 - 3a = -2a
\][/tex]
4. Solve for [tex]\(a\)[/tex]:
Combine like terms:
[tex]\[
3 - 3a = -2a
\][/tex]
Add [tex]\(3a\)[/tex] to both sides to isolate the variable:
[tex]\[
3 = a
\][/tex]
Thus, the value of [tex]\(a\)[/tex] that satisfies the given equation is:
[tex]\[
a = 3
\][/tex]
After examining all the provided choices, we conclude that the correct answer is:
[tex]\(a = 3\)[/tex].
