Question

The greatest number among \( 2^{3000} \), \( 3^{2000} \), \( 4^{1000} \), \( 2^{1000} \times 3^{1000} \) is:

• A. \( 2^{3000} \)
• B. \( 2^{1000} \times 3^{1000} \)
• C. \( 3^{2000} \)
• D. \( 4^{1000} \)

Hint:

To compare large numbers with exponents, take the logarithms of the numbers to convert them into simpler forms and compare the values.

Answer 1

The Correct Option is A

Given: \( 2^{300} \), \( 3^{200} \), \( 4^{100} \), \( 2^{100} + 3^{100} \) Goal: Find the greatest number. Steps:

1. Rewrite numbers:
   • \( 2^{300} = (2^3)^{100} = 8^{100} \)
   • \( 3^{200} = (3^2)^{100} = 9^{100} \)
   • \( 4^{100} \)
   • \( 2^{100} + 3^{100} \)
2. Compare:
   • \( 8^{100} \), \( 9^{100} \), \( 4^{100} \) are easy to compare: \( 9^{100} > 8^{100} > 4^{100} \)
   • \( 2^{100} + 3^{100} \): We need to compare it to \( 9^{100} \).
3. Estimate:
   • \( (2^{100} + 3^{100})^2 < (3^{100} + 3^{100})^2 = (2 \cdot 3^{100})^2 = 4 \cdot (3^{200}) = 4 \cdot 9^{100} \)
   • \( 2^{100} + 3^{100} < 2 \cdot 3^{100} = 2 \cdot 9^{50} \)
4. Compare \( 9^{100} \) and \( 2^{100} + 3^{100} \):
   • Consider \( \frac{2^{100} + 3^{100}}{9^{100}} = \left( \frac{2}{9} \right)^{100} + \left( \frac{1}{3} \right)^{100} \)
   • \( \left( \frac{2}{9} \right)^{100} \) is very small. \( \left( \frac{1}{3} \right)^{100} \) is very small. Their sum is very small
5. Conclusion: \( 9^{100} \) is the largest.

Answer: (a) \( 3^{200} \)