Which number is higher $2^{600}$ or $3^{400}$?

Question

Which number is higher $2^{600}$ or $3^{400}$ ?

I know that the solution is $3^{400}>2^{600}$ bot how to explain that. without using a calculator.

Answer 1

Hint $$3^2>2^3\Longrightarrow 3^4>2^6\Longrightarrow 3^{400}>2^{600}$$

Answer 2

Hint

Compare $2^6$ and $3^4$ and notice that the function $x\mapsto x^{100}$ is strictly increasing.

Answer 3

Another suggestion - apply $\sqrt[200]{\dots}$ on each one of the expressions:

• $\sqrt[200]{2^{600}}=2^{\frac{600}{200}}=2^3=8$
• $\sqrt[200]{3^{400}}=3^{\frac{400}{200}}=3^2=9$