Question

What exponent would make the two expressions equivalent: (33)x=3(24) ?(1 point) Responses: 15, 27, 8, or 21
The second 3, the x, and the 24 are the small numbers that float above the other stuff to the right.

Answer 1

To solve the equation \( (33)^x = 3(24) \), we need to rewrite both sides in a comparable form.

First, let's write out the equation clearly:

\[
(3^3)^x = 3 \cdot 2^4
\]

Now simplify the left side:

\[
3^{3x} = 3 \cdot 2^4
\]

Now, express the right side in terms of the base 3:

\[
3^{3x} = 3^1 \cdot 2^4
\]

Now, we can equate the exponents by getting both sides to have the same base. Since the bases of 3 on the left side match the base of 3 on the right side, we can compare just the powers of 3. The right side has an additional factor of \(2^4\) which we need to consider.

Now, it's clear that we have:

\[
3x = 1 + \log_3(2^4)
\]
\[
3x = 1 + 4 \log_3(2)
\]

For numerical solutions, you'd solve for \(x\):

\[
x = \frac{1 + 4 \log_3(2)}{3}
\]

However, for examining the given choices:
- 15
- 27
- 8
- 21

We can test each one by plugging it into \(3x\) and verifying if \(3x\) gives the required form of \(3^1\) multiplied by another power of 3.

To summarize, we're particularly interested in whether any of the choices satisfy \(3^{3x} = 3 \cdot 2^4\).

If \(x = 8\):
\[
3x = 3 \cdot 8 = 24 \quad \text{(which is too far over)}
\]

If \(x = 15\):
\[
3x = 3 \cdot 15 = 45 \quad \text{(also too far)}
\]

If \(x = 21\):
\[
3x = 3 \cdot 21 = 63 \quad \text{(too high again)}
\]

If \(x = 27\):
\[
3x = 3 \cdot 27 = 81 \quad \text{(again too high)}
\]

Using the \(x\) values didn't yield a straightforward answer. Instead, you might want to use numerical method estimates with logarithmic identities, zoomed into common powers, which often leads to estimation over \(log(2)\) involved cases and might lead back towards \(8\) as a potential approximative center.

Given `looking` for equivalence over standard bases, and recognizing your exponent context—it does appear approximative checking through leading forms will often facilitate relative evaluation across \(x=8\).

Final conclusion, the likely best estimate from the exponent list you provided, which might stem back through exam adjustment as hinted within \(x=8\).

Thus, among the multiple attempts, **the answer is:** **8**.