To solve the equation [tex]\( 7^{3x - 8} = \frac{1}{49} \)[/tex], follow these detailed steps:
1. Rewrite [tex]\( \frac{1}{49} \)[/tex] as a power of 7:
[tex]\[
\frac{1}{49} = 49^{-1}
\][/tex]
Since [tex]\( 49 = 7^2 \)[/tex], we can write:
[tex]\[
49^{-1} = \left( 7^2 \right)^{-1} = 7^{-2}
\][/tex]
Therefore, the original equation can be rewritten as:
[tex]\[
7^{3x - 8} = 7^{-2}
\][/tex]
2. Since the bases are the same, set the exponents equal to each other:
[tex]\[
3x - 8 = -2
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
3x - 8 = -2
\][/tex]
Add 8 to both sides:
[tex]\[
3x - 8 + 8 = -2 + 8
\][/tex]
Simplifying this gives:
[tex]\[
3x = 6
\][/tex]
Divide both sides by 3:
[tex]\[
x = \frac{6}{3}
\][/tex]
Simplifying further:
[tex]\[
x = 2
\][/tex]
Therefore, the solution to the equation [tex]\( 7^{3x - 8} = \frac{1}{49} \)[/tex] is [tex]\( x = 2 \)[/tex].
