Question

What exponent would make the two expressions equivalent: (33)x=324 ?(1 point) Responses 8 8 21 21 27 27 15

Answer 1

To solve for \( x \) in the equation \( (33)^x = 324 \), we can start by expressing 324 in terms of powers of 33.

First, let's take the logarithm of both sides:

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

Now, we can solve for \( x \):

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

Now, let's calculate the values:

1. \( \log(324) \) can be calculated directly:
- \( 324 = 18^2 = (3^2 \cdot 2)^2 = 3^4 \cdot 2^2 \)
- Thus, \( \log(324) = 2 \log(18) = 2 \log(2 \cdot 3^2) = 2 (\log(2) + 2 \log(3)) \)

2. For \( \log(33) \):
- \( 33 = 3 \cdot 11 \)
- \( \log(33) = \log(3) + \log(11) \)

Now let’s approximate values for logarithms using a calculator or logarithm tables:

- \( \log(2) \approx 0.301 \)
- \( \log(3) \approx 0.477 \)
- \( \log(11) \approx 1.041 \)

Calculating \( \log(324) \):

\[
\log(324) \approx 2(0.301 + 2 \times 0.477) = 2(0.301 + 0.954) \approx 2(1.255) \approx 2.51
\]

Calculating \( \log(33) \):

\[
\log(33) \approx 0.477 + 1.041 \approx 1.518
\]

Now plug into the formula for \( x \):

\[
x \approx \frac{2.51}{1.518} \approx 1.65
\]

This doesn't directly relate to the options provided, however, we can see if substitutive values for the provided options work.

Testing the options:
1. For \( x = 8: (33^8) \)
2. For \( x = 21: (33^{21}) \)
3. For \( x = 27: (33^{27}) \)
4. For \( x = 15: (33^{15}) \)

Plugging these into a calculator to check approximation to 324 confirms none of them fit.

Ultimately, \( (33)^x = 324 \) does not yield an easy direct exponent. However, addressing logical reasoning through inference, both values \( \log(33) \) and \( \log(324) \) do not correspond neatly to any of the choices rapidly.

If guessing was the only strategy, then \( x=8 \) may more closely connect to expectations for chunk size.

Thus:

The most suitably guessed exponent would be \( \textbf{8} \).

Please check accuracy via computing or logarithm means for final resolve.