(a) compute \(10^{999}\cdot 5^{-998}\cdot 2^{-997}\).
(b) The number \(12^{10}\cdot 6^{-8}\) is an integer. How many digits does it have?
(c) What is the sum of all positive integers smaller than 1000 that can be written in the form \(100\cdot 2^n\), where \(n\) is an integer (not necessarily positive)?
+538 a) 5^x*2^x=10^x
so 5^-997*2&-997=10^-997
10^-997*10^999=10^2 or 100
Extra 5: 100/5=20
b) 12^10= 2^20*3^10
6^-8= 2^-8*3^-8
2^20*2^-8=2^12=4096
3^10*3^-8=3^2 OR 9
4096*9=36864
There are 5 digits
c)
-2, -1, 0, 1, 2, 3
25, 50, 100, 200, 400, 800
sum is 1575
