You will get two equations of the form $12 = k\cdot 3^{2m}$ and $324 = k\cdot 3^{5m}$. How can dividing one equation by the other help you eliminate $k$ and relate only powers of $3$?

After you divide, you should get an equation like $3^{3m} = 27$. Rewrite $27$ as a power of $3$ so you can set the exponents equal and solve for $m$.

Once you know $m$, plug it back in to find $k$, then substitute both $k$ and $m$ into $p(8) = k\bigl(3^{8m}\bigr)$ and simplify carefully using exponent rules.

In Desmos, type 324/12 and press Enter. This computes $\dfrac{p(5)}{p(2)}$, the factor by which the function grows when $x$ increases from $2$ to $5$.

Now type 27^(1/3) (using the result from the previous step). This gives the factor by which $p(x)$ grows when $x$ increases by 1 (the cube root of the 3-step growth).

From $x=2$ to $x=8$ is 6 steps of size 1, so in Desmos type 12*3^6 (starting value at $x=2$ times the 1-step growth factor raised to the 6-step difference). The value shown is $p(8)$; match this number to the correct answer choice.

The function is

Use the two given values:

• $p(2)=12$ gives $12 = k\bigl(3^{2m}\bigr)$.
• $p(5)=324$ gives $324 = k\bigl(3^{5m}\bigr)$.

So we have the system:

Divide the second equation by the first to cancel $k$:

The $k$ cancels and exponents subtract:

Compute the left side: $324/12 = 27$, so

Since $27 = 3^3$, we have $3^{3m} = 3^3$, so $3m = 3$ and therefore $m = 1$.

Substitute $m=1$ into $12 = k\bigl(3^{2m}\bigr)$:

So

Now substitute $k=\tfrac{4}{3}$ and $m=1$ into $p(x) = k\bigl(3^{mx}\bigr)$ with $x=8$:

Compute $3^8$:

• $3^4 = 81$,
• $3^8 = (3^4)^2 = 81^2 = 6561$.

So

Thus $p(8) = 8{,}748$, which corresponds to choice B) $8\,748$.