Question

Solve for

\[x\].

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

Answer 1

[AI-generated answer — Bot GPT 3.5]

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 2

wrong

Answer 3

[AI-generated answer — Bot GPT 3.5]

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 $x = \boxed{5}$.

Answer 4

[AI-generated answer — Explain Bot]

To solve for \(x\) in the equation \(\frac{11^x}{11^3} = 11^2\), we can start by simplifying the expression on the left-hand side of the equation.

We know that \(a^{m} \cdot a^{n} = a^{m+n}\) for any positive real numbers \(a\) and any real numbers \(m\) and \(n\).

Using this property, we can rewrite the left-hand side of the equation as:

\(\frac{11^x}{11^3} = 11^{x-3}\)

So now our equation becomes:

\(11^{x-3} = 11^2\)

Since the bases on both sides of the equation are equal (both are 11), the exponents must also be equal. Therefore, we can set \(x-3\) equal to \(2\) and solve for \(x\):

\(x - 3 = 2\)

Adding 3 to both sides of the equation, we get:

\(x = 5\)

Therefore, the solution to the equation \(\frac{11^x}{11^3} = 11^2\) is \(x = 5\).