Question

solve. write your answer as an integer or a fraction in simplest form.
$9^{3x + 1} = 3$
$x = $

Explanation:

Step1: Express 9 as a power of 3

Since \( 9 = 3^2 \), we can rewrite the left side of the equation \( 9^{3x + 1} = 3 \) as \( (3^2)^{3x + 1} \). Using the exponent rule \( (a^m)^n=a^{mn} \), we get \( 3^{2(3x + 1)} \).

Step2: Simplify the exponent

Simplify \( 2(3x + 1) \) to \( 6x + 2 \). So the equation becomes \( 3^{6x + 2}=3^1 \).

Step3: Set exponents equal

Since the bases are the same (both are 3), we can set the exponents equal to each other: \( 6x + 2 = 1 \).

Step4: Solve for x

Subtract 2 from both sides: \( 6x = 1 - 2=-1 \). Then divide both sides by 6: \( x = -\frac{1}{6} \).

Answer:

\( -\frac{1}{6} \)