Which number is larger `(1.3)^(2000) or 600`

Answer

To determine which number is larger between \( (1.3)^{2000} \) and \( 600 \), we can use the binomial theorem to approximate \( (1.3)^{2000} \).

### Step-by-step Solution:

1. **Rewrite the Expression**:
\[ (1.3)^{2000} = (1 + 0.3)^{2000} \]

2. **Apply the Binomial Theorem**:
The binomial theorem states that:
\[ (1 + x)^n \approx 1 + nx \quad \text{for small } x \]
Here, \( x = 0.3 \) and \( n = 2000 \).

3. **Calculate the Approximation**:
Using the first two terms of the binomial expansion:
\[ (1 + 0.3)^{2000} \approx 1 + 2000 \times 0.3 \]
Calculate \( 2000 \times 0.3 \):
\[ 2000 \times 0.3 = 600 \]
Thus, we have:
\[ (1.3)^{2000} \approx 1 + 600 = 601 \]

4. **Compare the Values**:
Now we compare \( 601 \) with \( 600 \):
\[ 601 > 600 \]

5. **Conclusion**:
Therefore, we conclude that:
\[ (1.3)^{2000} > 600 \]

### Final Answer:
The number \( (1.3)^{2000} \) is larger than \( 600 \).