Question

solve. write your answer as an integer or a fraction in simplest form.
$36^{x + 3} = 6$
$x = \boxed{\dfrac{\\,\\,}{\\,\\,}}$

Explanation:

Step1: Express 36 as a power of 6

Since \( 36 = 6^2 \), we can rewrite the left - hand side of the equation \( 36^{x + 3}=6 \) as \( (6^2)^{x + 3} \).
According to the power - of - a - power rule \((a^m)^n=a^{m\times n}\), we have \((6^2)^{x + 3}=6^{2(x + 3)}\). So the equation becomes \( 6^{2(x + 3)}=6^1 \).

Step2: Set the exponents equal

If two exponential expressions with the same base (\(a^m=a^n\) where \(a>0,a
eq1\)) are equal, then their exponents are equal. So we can set up the equation \( 2(x + 3)=1 \).

Step3: Solve for x

First, expand the left - hand side: \( 2x+6 = 1 \).
Then, subtract 6 from both sides: \( 2x=1 - 6=-5 \).
Finally, divide both sides by 2: \( x=-\frac{5}{2} \).

Answer:

\(x = -\frac{5}{2}\)