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(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)?

 Jan 10, 2021
 #1
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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

 Jan 10, 2021

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