To solve the equation [tex]\(3^{2x + 1} = 3^{x + 5}\)[/tex], we can make use of the property of exponents that if the bases are the same, the exponents must also be equal. Here are the steps:
1. Set the Exponents Equal to Each Other:
Since the bases [tex]\(3\)[/tex] on both sides of the equation are the same, we can set the exponents equal to each other:
[tex]\[
2x + 1 = x + 5
\][/tex]
2. Isolate the Variable [tex]\(x\)[/tex]:
To find the value of [tex]\(x\)[/tex], we need to isolate [tex]\(x\)[/tex] on one side of the equation. Start by subtracting [tex]\(x\)[/tex] from both sides of the equation:
[tex]\[
2x + 1 - x = x + 5 - x
\][/tex]
Simplify this to:
[tex]\[
x + 1 = 5
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Next, we isolate [tex]\(x\)[/tex] by subtracting [tex]\(1\)[/tex] from both sides of the equation:
[tex]\[
x + 1 - 1 = 5 - 1
\][/tex]
Simplify this to:
[tex]\[
x = 4
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] is [tex]\(4\)[/tex]. The correct answer is [tex]\(4\)[/tex].
