the smallest natural number n that satisfies 12^200<n^300

To find the smallest natural number \( n \) that satisfies the inequality \( 12^{200} < n^{300} \), let's follow these steps:

1. **Rewrite the inequality**:
   \[
   12^{200} < n^{300}
   \]

2. **Take the 300th root of both sides** to solve for \( n \):
   \[
   \left(12^{200}\right)^{\frac{1}{300}} < \left(n^{300}\right)^{\frac{1}{300}}
   \]
   Simplifying both sides:
   \[
   12^{\frac{200}{300}} < n
   \]
   \[
   12^{\frac{2}{3}} < n
   \]

3. **Simplify the exponent** \( \frac{2}{3} \):
   \[
   12^{\frac{2}{3}} = \left(12^2\right)^{\frac{1}{3}} = 144^{\frac{1}{3}}
   \]

4. **Approximate** \( 144^{\frac{1}{3}} \):
   The cube root of 144 is approximately \( 5.24 \).

Thus, the smallest integer \( n \) that satisfies \( n > 5.24 \) is \( n = 6 \).

So, the smallest natural number \( n \) that satisfies \( 12^{200} < n^{300} \) is \( \boxed{6} \).

Would you like any details or have any questions? 

Here are 5 related questions for further exploration:

1. What is the smallest \( n \) that satisfies \( 8^{150} < n^{250} \)?
2. How would the solution change if the exponent on \( n \) was 400 instead of 300?
3. Can you generalize a formula for \( n \) given \( a^b < n^c \)?
4. What happens if we change the base 12 to 16 in the original problem?
5. How would you approach this problem if it were given with non-integer exponents?

**Tip:** When dealing with inequalities involving exponents, taking roots is a common technique to simplify and solve the problem.

Learn how to find the smallest natural number n that satisfies the inequality 12^200 < n^300 using mathematical techniques. Follow a step-by-step solution to understand how exponents and roots can be used to solve such inequalities.

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