Question

Solve for
\[x\].

\[\dfrac{11^{x}}{11^3}=11^2\]

Answer 1

wrong

Answer 2

We have \[\dfrac{11^{x}}{11^3}=(11^1)^2=11^2.\]
Because $11^3$ is a nonzero number, we can divide both sides by $11^3$ to obtain
\[\frac{11^x}{11^3}=\frac{11^2}{11^3}.\]
Then we have
\[11^{x-3}=\frac{1}{11}.\]
Because $11^3$ is a nonzero number, we can multiply both sides by $11^3$ to get
\[11^{x-3}\cdot11^3=1.\]
Using the property of exponentiation $a^{m+n}=a^ma^n$ and $a^{-n}=\frac{1}{a^n}$, we have
\[11^{x-3+3}=1^3=1.\]
Thus, we have $11^{x}=1$.

Since $11^0=1$, we have $x=\boxed{0}.$

Answer 3

Apologies for the error. Let's try again. We have
\[\frac{11^x}{11^3} = 11^2.\]Since the base is the same on both sides, we can subtract the exponents:
\[x - 3 = 2.\]Adding $3$ to both sides gives $x = \boxed{5}$.