To solve the equation [tex]\(\left(\frac{1}{9}\right)^2 = 81^{2+1} \cdot 27^{2-a}\)[/tex], we begin by rewriting the bases in terms of powers of 3.
1. Rewriting both sides of the equation:
[tex]\[
\left(\frac{1}{9}\right)^2 = \left(\frac{1}{3^2}\right)^2 = 3^{-4}
\][/tex]
[tex]\[
81^{2+1} \cdot 27^{2-a} = 81^3 \cdot 27^{2-a}
\][/tex]
2. Expressing 81 and 27 as powers of 3:
- [tex]\(81 = 3^4\)[/tex]
- [tex]\(27 = 3^3\)[/tex]
3. Substitute these powers back into the equation:
[tex]\[
3^{-4} = (3^4)^3 \cdot (3^3)^{2-a}
\][/tex]
4. Simplify the exponents in the right-hand side expression:
[tex]\[
(3^4)^3 = 3^{4 \cdot 3} = 3^{12}
\][/tex]
[tex]\[
(3^3)^{2-a} = 3^{3 \cdot (2-a)} = 3^{6 - 3a}
\][/tex]
So the equation becomes:
[tex]\[
3^{-4} = 3^{12} \cdot 3^{6 - 3a}
\][/tex]
5. Combine the exponents on the right-hand side:
[tex]\[
3^{-4} = 3^{12 + 6 - 3a} = 3^{18 - 3a}
\][/tex]
6. Since the bases are the same, we can equate the exponents:
[tex]\[
-4 = 18 - 3a
\][/tex]
7. Solve for [tex]\(a\)[/tex]:
[tex]\[
-4 = 18 - 3a
\][/tex]
Subtract 18 from both sides:
[tex]\[
-4 - 18 = -3a
\][/tex]
[tex]\[
-22 = -3a
\][/tex]
Divide both sides by -3:
[tex]\[
a = \frac{22}{3} \approx 7.333
\][/tex]
Thus, the value of [tex]\(a\)[/tex] is approximately [tex]\(7.333\)[/tex].
So the value of [tex]\(a\)[/tex] is not listed among the given choices ([tex]\(-4, -2, 2, 6\)[/tex]). Therefore, there seems to be a misunderstanding, but the correct mathematical value is [tex]\( \boxed{ \approx 7.333 } \)[/tex].
