To solve the equation [tex]\( 16 = 2^{3x + 1} \)[/tex] for [tex]\( x \)[/tex], we can proceed with the following steps:
1. Recognize the base structure: Notice that 16 can be expressed as a power of 2. Specifically, [tex]\( 16 = 2^4 \)[/tex].
2. Rewrite the equation: Substitute [tex]\( 16 \)[/tex] with [tex]\( 2^4 \)[/tex]. Thus, the equation becomes:
[tex]\[
2^4 = 2^{3x + 1}
\][/tex]
3. Set the exponents equal: Since the bases are the same (both are base 2), we can set the exponents equal to each other:
[tex]\[
4 = 3x + 1
\][/tex]
4. Solve for [tex]\( x \)[/tex]: Isolate [tex]\( x \)[/tex] by following these steps:
- Subtract 1 from both sides to get:
[tex]\[
4 - 1 = 3x
\][/tex]
Simplifying this, we get:
[tex]\[
3 = 3x
\][/tex]
- Divide both sides by 3 to solve for [tex]\( x \)[/tex]:
[tex]\[
\frac{3}{3} = x
\][/tex]
Simplifying the division:
[tex]\[
x = 1
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is [tex]\( \boxed{1} \)[/tex].
