Great Power
Which is greater: $10^{250}$ or $6^{300}$?
Which is greater: $10^{250}$ or $6^{300}$?
This problem is adapted from the World Mathematics Championships
Answer: $10^{250}$
$10^{250}$ or $6^{300}$
$2^{250}\times5^{250}$ or $2^{300}\times3^{300}$
$5^{250}$ or $2^{50}\times3^{300}$
$=6^{50}\times3^{250}$
$\left(5^5\right)^{50}$ or $\left(6\times3^5\right)^{50}$
$5^5$ or $6\times3^5$
$\left(\frac53\right)^5$ or $6$
$\underbrace{\frac{25}9}_{\gt2}\times\underbrace{\frac{25}9}_{\gt2}\times\underbrace{\frac53}_{\gt1.5}$ or $6$
$2\times2\times1.5= 6$ so $10^{250}$ is greater