How many digits are in the decimal representation of 5^40?

No calculator allowed.

log(5^40) = 40log5 = 27.95

So, the number is between 10^27 and 10^28

so, ...

To determine how many digits are in the decimal representation of 5^40, we need to examine the number itself.

We know that any number raised to the power of 0 is equal to 1, so 5^0 = 1.

To find subsequent powers of 5, we can observe a pattern:
5^1 = 5
5^2 = 5 * 5 = 25
5^3 = 5 * 5 * 5 = 125
5^4 = 5 * 5 * 5 * 5 = 625

We can notice that as the exponent increases, the number of digits also increases. Therefore, we can conclude that 5^40 will have more than 2 digits.

To estimate the number of digits more accurately, we can look at the number of digits in powers of 10:
10^0 = 1
10^1 = 10
10^2 = 100
10^3 = 1,000
10^4 = 10,000

We can observe that as the power of 10 increases, the number of digits also increases. Based on this observation, we can infer that 5^40 will have less than the number of digits in 10^40, which is 41 digits.

Therefore, the decimal representation of 5^40 will have either 40 or 41 digits.