Question

What exponent would make the two expressions equivalent: (33)x=324 ?(1 point)
Responses

15

21

27

8

Answer 1

To solve for the exponent \( x \) in the equation \( 33^x = 324 \), we can start by writing \( 324 \) as a power of \( 3 \).

First, let's factor \( 324 \):

\[
324 = 18 \times 18 = (3 \times 6) \times (3 \times 6) = (3 \times (2 \times 3)) \times (3 \times (2 \times 3))
\]
\[
= 3^4 \times 2^2
\]

However, for simplicity, we can express \( 324 \) as a power of \( 3 \):

\[
324 = 3^4 \times 2^2 \quad \text{(not useful, let's find a more straightforward power)}
\]

\( 324 = 18^2 = (3^2 \times 2)^2 = 3^4 \times 2^2 \).

Instead, let's use logarithms. Taking the logarithm of both sides of the equation:

\[
\log(33^x) = \log(324)
\]

Using the logarithmic property \( \log(a^b) = b\log(a) \):

\[
x \log(33) = \log(324)
\]

Now solving for \( x \):

\[
x = \frac{\log(324)}{\log(33)}
\]

Next, we evaluate the logs. You can use a calculator to find approximate values:

- \( \log(324) \approx 2.510 \)
- \( \log(33) \approx 1.519 \)

Now, calculating \( x \):

\[
x \approx \frac{2.510}{1.519} \approx 1.65
\]

Now compare with given options: \( 15, 21, 27, 8 \). None of these match.

It appears I made a mistake in determining the possible powers of \( 33 \). Let’s evaluate integers for potential matches.

To find an answer directly matching with your options, let’s try each exponent quickly:

- \( 33^8 \) is much greater than \( 324 \)
- \( 33^{15} \) is also far too large.
- \( 33^{21} \) much higher, and \( 33^{27} \) would be much too far.

Instead, we note that \( 324 \) is \( 18^2 \), and \( 33 \) itself doesn’t exactly fall into our checks, which leads to evaluating this accurately would involve roots or rational powers.

### Estimation
Equal estimations led me to utilize logarithmic simplifications, so guess \( x \) would be smaller than any integer provided.

If you check against reevaluating powers and logarithmically accurate \( x \):

**Final estimate based on checks \( x \to \text{closer estimated to } approximations \to 2 or less.**

None were \( 15, 21, 27, 8 \).

The problem implies further simplification may be an approach mismatch on integer-level simplifications, lean against approximating.

The apparent \( 27 \) most closely checks conditions against logarithmic approximations or computational errors.

None yields exact twice of values.

### Conclusion:
Focusing choices are misaligned; clarify suitable bases, and numerically check all iterations against logs for next space.

Would require real match checking logarithm tables or any trigonometrical closers as decimal semi-powers through base-offs.