To solve the equation [tex]\( 3^{a-18} = \frac{1}{81^{2a}} \)[/tex], we can follow these steps:
1. Rewrite the equation with the same base:
Notice that 81 can be written as a power of 3:
[tex]\[
81 = 3^4
\][/tex]
Therefore, the equation [tex]\(\frac{1}{81^{2a}}\)[/tex] can be rewritten as:
[tex]\[
\frac{1}{(3^4)^{2a}}
\][/tex]
2. Simplify the exponent:
Using the property [tex]\((x^m)^n = x^{mn}\)[/tex], we get:
[tex]\[
(3^4)^{2a} = 3^{4 \cdot 2a} = 3^{8a}
\][/tex]
So, the equation becomes:
[tex]\[
\frac{1}{3^{8a}}
\][/tex]
3. Use reciprocal property:
Recall that [tex]\( \frac{1}{x^n} = x^{-n} \)[/tex], therefore:
[tex]\[
\frac{1}{3^{8a}} = 3^{-8a}
\][/tex]
Now, the equation looks like:
[tex]\[
3^{a-18} = 3^{-8a}
\][/tex]
4. Equate the exponents:
Since the bases are the same (both are base 3), we can set their exponents equal to each other:
[tex]\[
a - 18 = -8a
\][/tex]
5. Solve for [tex]\(a\)[/tex]:
Add [tex]\(8a\)[/tex] to both sides of the equation to collect terms involving [tex]\(a\)[/tex]:
[tex]\[
a + 8a - 18 = -8a + 8a
\][/tex]
Simplify:
[tex]\[
9a - 18 = 0
\][/tex]
6. Isolate [tex]\(a\)[/tex]:
Add 18 to both sides to isolate the term with [tex]\(a\)[/tex]:
[tex]\[
9a - 18 + 18 = 0 + 18
\][/tex]
Simplify:
[tex]\[
9a = 18
\][/tex]
7. Divide by 9:
[tex]\[
a = \frac{18}{9}
\][/tex]
Simplify:
[tex]\[
a = 2
\][/tex]
Therefore, the solution to the equation [tex]\( 3^{a-18} = \frac{1}{81^{2a}} \)[/tex] is [tex]\( a = 2 \)[/tex].
