5. Solve the equation $$ \frac{\left(3^{n} \frac{1}{2} ight.}{3^{\frac{1}{2}}=3 .} $$ $$ \begin{array}{ll} ext { (A) } x=3 & ext { (C) } x=\frac{49}{4} \ ext { (B) } x=-1 & ext { (D) } x=1\end{array} $$

Question

5. Solve the equation $$ \frac{\left(3^{n} \frac{1}{2}ight.}{3^{\frac{1}{2}}=3 .} $$ $$ \begin{array}{ll}	ext { (A) } x=3 & 	ext { (C) } x=\frac{49}{4} \ 	ext { (B) } x=-1 & 	ext { (D) } x=1\end{array} $$

UpStudy AI · Accepted Answer

Let's solve the equation step by step. The equation given is:

$$
\frac{3^{n} \cdot \frac{1}{2}}{3^{\frac{1}{2}}} = 3
$$

1. **Simplify the left side of the equation**:
   We can rewrite the left side by simplifying the fraction:

$$
   \frac{3^{n} \cdot \frac{1}{2}}{3^{\frac{1}{2}}} = \frac{3^{n}}{3^{\frac{1}{2}}} \cdot \frac{1}{2} = 3^{n - \frac{1}{2}} \cdot \frac{1}{2}
   $$

So the equation becomes:

$$
   3^{n - \frac{1}{2}} \cdot \frac{1}{2} = 3
   $$

2. **Multiply both sides by 2**:
   To eliminate the fraction, we multiply both sides by 2:

$$
   3^{n - \frac{1}{2}} = 6
   $$

3. **Take the logarithm of both sides**:
   We can take the logarithm (base 3) of both sides to solve for $$n$$:

$$
   n - \frac{1}{2} = \log_{3}(6)
   $$

4. **Solve for $$n$$**:
   Now, we can isolate $$n$$:

$$
   n = \log_{3}(6) + \frac{1}{2}
   $$

Next, we need to evaluate $$\log_{3}(6)$$. We can express 6 in terms of base 3:

$$
\log_{3}(6) = \log_{3}(2 \cdot 3) = \log_{3}(2) + \log_{3}(3) = \log_{3}(2) + 1
$$

Thus, we have:

$$
n = \log_{3}(2) + 1 + \frac{1}{2} = \log_{3}(2) + \frac{3}{2}
$$

Now, we can evaluate $$\log_{3}(2)$$ using a calculator or logarithm properties, but since we are looking for a numerical answer, we can approximate it.

Let's calculate $$\log_{3}(2)$$ and then find $$n$$.

Using the change of base formula:

$$
\log_{3}(2) = \frac{\log_{10}(2)}{\log_{10}(3)}
$$

Now, let's calculate this value.

I'll perform the calculations to find $$n$$.
Numbers by following steps:
- step0: Solution:
  $$1.5$$
Calculate or simplify the expression $$ \log(2)/\log(3) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\log_{10}{\left(2\right)}}{\log_{10}{\left(3\right)}}$$
- step1: Use the logarithm base change rule:
  $$\log_{3}{\left(2\right)}$$
We have calculated:

$$
\log_{3}(2) \approx 0.63093
$$

Now, substituting this value back into our expression for $$n$$:

$$
n = \log_{3}(2) + \frac{3}{2} \approx 0.63093 + 1.5
$$

Now, let's perform the addition:

$$
n \approx 0.63093 + 1.5 = 2.13093
$$

Thus, the approximate value of $$n$$ is:

$$
n \approx 2.13093
$$

Now, let's check the options provided:

- (A) $$x=3$$
- (B) $$x=-1$$
- (C) $$x=\frac{49}{4}$$
- (D) $$x=1$$

None of the options directly match $$n \approx 2.13093$$. However, we can convert $$\frac{49}{4}$$ to a decimal to see if it is close to our result:

$$
\frac{49}{4} = 12.25
$$

Since $$2.13093$$ does not match any of the options, it seems there might be a misunderstanding in the problem statement or the options provided.

If you have any further clarifications or adjustments to the problem, please let me know!
The value of $$ n $$ is approximately 2.13, which does not match any of the provided options.

5. Solve the equation \( \frac{\left(3^{n} \frac{1}{2}\right.}{3^{\frac{1}{2}}=3 .} \) \( \begin{array}{ll}\text { (A) } x=3 & \text { (C) } x=\frac{49}{4} \\ \text { (B) } x=-1 & \text { (D) } x=1\end{array} \)

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