Which of the following is/are true?

Options

A. `100^(300)lt300!`
B. `sqrt(300^(300))lt 300!`
C. `100^(300)gt300!`
D. `sqrt(300^(300))gt300!`

Explanation

To determine which of the given statements are true, we need to analyze the inequalities involving \(100^{300}\), \(300!\), and \(\sqrt{300^{300}}\).

### Step 1: Understanding the relationship between \(n!\) and \(n^n\)

We know from the properties of factorials and exponentials that:

\[ n! > \frac{n^n}{e^n} \]

This implies that:

\[ n! > \left(\frac{n}{e}\right)^n \]

For \(n = 300\):

\[ 300! > \left(\frac{300}{e}\right)^{300} \]

### Step 2: Approximating \(e\)

Since \(e \approx 2.718\), we can approximate it as \(3\) for simplicity in inequalities:

\[ 300! > \left(\frac{300}{3}\right)^{300} = 100^{300} \]

Thus, we conclude:

\[ 300! > 100^{300} \]

### Step 3: Evaluating the first statement

The first statement is:

\[ 100^{300} < 300! \]

From our previous conclusion, this statement is **true**.

### Step 4: Evaluating the second statement

Next, we need to evaluate:

\[ \sqrt{300^{300}} < 300! \]

We can simplify \(\sqrt{300^{300}}\):

\[ \sqrt{300^{300}} = 300^{150} \]

Now, we need to check if:

\[ 300^{150} < 300! \]

Using the earlier result \(300! > 100^{300}\), we can relate this to \(300^{150}\):

Since \(100^{300} = (10^2)^{300} = 10^{600}\) and \(300^{150} = (3 \times 100)^{150} = 3^{150} \times 100^{150}\), we can see that \(300^{150}\) is significantly smaller than \(300!\) due to the rapid growth of factorials.

Thus, we conclude:

\[ 300^{150} < 300! \]

### Step 5: Final conclusion

Both statements are true:

1. \(100^{300} < 300!\)
2. \(\sqrt{300^{300}} < 300!\)

### Summary of Findings

The true statements are:

- \(100^{300} < 300!\)
- \(\sqrt{300^{300}} < 300!\)