Comparing big powers

Question

Which of the following is the largest?

A. $1^{200}$

B. $2^{400}$

C.$4^{80}$

D. $6^{300}$

E. $10^{250}$

I'm stuck trying to solve this. Obviously A and C are wrong ($4^{80}$ is less than $2^{400}$ and 1 to any power is always 1). And cancelling $2^{200}$ from each of the remaining choices, I can also eliminate B. However, I don't really know how to compare D and E... Any hints or helps?

Answer 1

$$6^{300} = (6^6)^{50} \ \ ; \ \ 10^{250} = (10^5)^{50}$$ so it is enough to check what is larger between $6^6$ and $10^5$. Now, $$6^6 = (6^3)^2 = 216^2 < 300^2 = 90000 < 100000 = 10^5$$

Answer 2

$$2^{400} = 1024^{40} < 10000^{40} = 10^{160} < 10^{250}$$

See the other answers for $6^{300}$ and $10^{250}$.

Answer 3

$$10^{250}>6^{300}$$ or $$10^5>6^6$$ or $$5^5>2\cdot3^6$$ or $$3125>1458,$$ which is obvious.

Now, it's obvious that $10^{250}$ is a largest number.

Answer 4

B is $2^{400}$

D is $2^{300}3^{300} \gt 2^{600}$

E is $10^{250} =2^{250}5^{250}$

Then we have $3^6=729, 5^5=3125\gt2^23^6$

So E is $10^{250} =2^{250}5^{250}\gt2^{250}(2^23^6)^{50}=2^{350}3^{300}\gt 2^{300}3^{300}$

Answer 5

$$ 10^{250}>6^{300} $$ $$ 10^{5}>6^{6} $$ $$ 5^{5} 2^{5}>2^{6} 3^{6}$$ $$ 5^{5} >2.3^{6}$$

Note that $ 3^2=9<10=5.2 $ so:

$$ 2.3^{6} < 2.10.10.10=2.10^3=2^4.5^3 $$ $$ 2.3^{6} < 2^4.5^3 < 5^5 $$ $$ 2^4 < 5^2 $$ $$ 4^2 < 5^2 $$