To solve the equation [tex]\( 27 = 3^{4x + 1} \)[/tex], we start by expressing 27 as a power of 3. We know that [tex]\( 27 = 3^3 \)[/tex]. So, we can rewrite the equation as:
[tex]\[ 3^3 = 3^{4x + 1} \][/tex]
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[ 3 = 4x + 1 \][/tex]
Next, solve for [tex]\( x \)[/tex] by isolating [tex]\( x \)[/tex] on one side of the equation. Subtract 1 from both sides:
[tex]\[ 3 - 1 = 4x \][/tex]
[tex]\[ 2 = 4x \][/tex]
Finally, divide both sides by 4 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{2}{4} \][/tex]
[tex]\[ x = \frac{1}{2} \][/tex]
Therefore, the solution to the equation [tex]\( 27 = 3^{4x + 1} \)[/tex] is [tex]\( x = \frac{1}{2} \)[/tex].
The correct answer is:
[tex]\[ \frac{1}{2} \][/tex]
