To solve the equation [tex]\(3^{2x - 1} = 9^{\frac{-x}{2}}\)[/tex], let’s break it down step by step:
1. Rewrite the bases to be the same:
The base of the exponent on the right side, 9, can be rewritten in terms of 3. Since [tex]\(9 = 3^2\)[/tex], the equation becomes:
[tex]\[
9^{\frac{-x}{2}} = (3^2)^{\frac{-x}{2}}
\][/tex]
2. Simplify the exponent:
Using the property of exponents [tex]\((a^m)^n = a^{mn}\)[/tex], we can simplify the right-hand side:
[tex]\[
(3^2)^{\frac{-x}{2}} = 3^{2 \cdot \frac{-x}{2}} = 3^{-x}
\][/tex]
Now the equation is:
[tex]\[
3^{2x - 1} = 3^{-x}
\][/tex]
3. Set the exponents equal to each other:
Since the bases are now the same, we can set the exponents equal to each other:
[tex]\[
2x - 1 = -x
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
To find [tex]\( x \)[/tex], combine like terms (i.e., get all [tex]\( x \)[/tex]-terms on one side of the equation):
[tex]\[
2x + x - 1 = 0
\][/tex]
Simplify:
[tex]\[
3x - 1 = 0
\][/tex]
Add 1 to both sides:
[tex]\[
3x = 1
\][/tex]
Divide both sides by 3:
[tex]\[
x = \frac{1}{3}
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is:
[tex]\[
x = \frac{1}{3}
\][/tex]
Converting [tex]\(\frac{1}{3}\)[/tex] to a decimal, we get approximately [tex]\( 0.3333333333333333 \)[/tex].
