which is greater, 100^10 or 10^100? explain reasoning.
10^100 = 10^2*10^98 = 100*10^98
Does that give you a start?
To determine which is greater between 100^10 and 10^100, let's first analyze the exponential expressions.
100^10 means taking 100 and multiplying it by itself 10 times:
100^10 = 100 * 100 * 100 * 100 * 100 * 100 * 100 * 100 * 100 * 100
10^100 means taking 10 and multiplying it by itself 100 times:
10^100 = 10 * 10 * 10 * ... * 10 (100 times)
Now, let's compare the number of terms in each expression:
In 100^10, there are 10 terms being multiplied together: 100 * 100 * 100 * 100 * 100 * 100 * 100 * 100 * 100 * 100.
In 10^100, there are 100 terms being multiplied together: 10 * 10 * 10 * ... * 10 (100 times).
Here's the key observation: 100 is a larger base than 10. When multiplying a number by itself multiple times, a larger base will result in a greater value.
Therefore, each term in 100^10 (100 * 100 * 100 * ... * 100) is larger than each term in 10^100 (10 * 10 * 10 * ... * 10), as we have 100 in each term in the former and only 10 in each term in the latter.
Since 100^10 has fewer terms but each term is larger, it will result in a greater value compared to 10^100 which has more terms but each term is smaller.
Hence, 100^10 is greater than 10^100.