To solve the equation [tex]\( 625 = 5^{6 - 2x} \)[/tex] for the variable [tex]\( x \)[/tex], we will proceed with the following steps:
1. Express both sides with the same base: \\
Notice that 625 can be written as a power of 5:
[tex]\[
625 = 5^4
\][/tex]
Therefore, the equation becomes:
[tex]\[
5^4 = 5^{6 - 2x}
\][/tex]
2. Set the exponents equal to each other: \\
Since the bases are the same and the exponents must be equal for the equation to hold true, we set the exponents equal:
[tex]\[
4 = 6 - 2x
\][/tex]
3. Solve for [tex]\( x \)[/tex]: \\
To isolate [tex]\( x \)[/tex], we perform the following steps:
[tex]\[
4 = 6 - 2x
\][/tex]
Subtract 6 from both sides:
[tex]\[
4 - 6 = -2x
\][/tex]
Simplify:
[tex]\[
-2 = -2x
\][/tex]
Divide both sides by -2:
[tex]\[
x = 1
\][/tex]
4. Conclusion:
The value of [tex]\( x \)[/tex] that satisfies the equation [tex]\( 625 = 5^{6 - 2x} \)[/tex] is [tex]\( \boxed{1} \)[/tex].
