Alright class, let's solve the equation [tex]\(3^{x+4} = \frac{1}{3^x}\)[/tex] step by step.
1. Rewrite the equation with a negative exponent: We know that [tex]\( \frac{1}{3^x} \)[/tex] can be expressed as [tex]\( 3^{-x} \)[/tex].
So, the equation becomes:
[tex]\[
3^{x+4} = 3^{-x}
\][/tex]
2. Equate the exponents: Since the bases (3) on both sides of the equation are the same, we can set the exponents equal to each other.
[tex]\[
x + 4 = -x
\][/tex]
3. Combine like terms: Let's bring all the [tex]\(x\)[/tex] terms to one side of the equation.
[tex]\[
x + 4 = -x
\][/tex]
Add [tex]\(x\)[/tex] to both sides:
[tex]\[
x + x + 4 = 0
\][/tex]
Combine the [tex]\(x\)[/tex] terms:
[tex]\[
2x + 4 = 0
\][/tex]
4. Isolate [tex]\(x\)[/tex]: Subtract 4 from both sides to begin isolating [tex]\(x\)[/tex].
[tex]\[
2x = -4
\][/tex]
5. Solve for [tex]\(x\)[/tex]: Divide both sides by 2 to solve for [tex]\(x\)[/tex].
[tex]\[
x = \frac{-4}{2}
\][/tex]
6. Simplify the result:
[tex]\[
x = -2
\][/tex]
So, the solution to the equation [tex]\(3^{x+4} = \frac{1}{3^x}\)[/tex] is [tex]\( \boxed{-2} \)[/tex].
