On the planet Zilot, the units of measurement are quite different from what we know on Earth. The standard unit of length is the Zimeter, and one of its submultiples is the Zimimimeter, which is equivalent to $10^{-7}$ Zimeters. A calculator can perform only two operations: multiplying a number by $10^{8}$ or dividing it by $10^{5}$. For example, using the operations of the calculator, we can perform the following conversions:

$$
3 \rightarrow 3 \cdot 10^{-5} \rightarrow 3 \cdot 10^{-10} \rightarrow 3 \cdot 10^{-2}
$$

a) Explain how we can combine the two operations of the calculator to make the number representing the conversion of $7 \cdot 10^{2}$ Zimeters to Zimimimeters appear on the screen.

b) How can we obtain the conversion of $10^{10} \cdot 10^{-4}$ Zimeters to Zimimimeters starting with the number 1000 on the calculator screen?

c) Using the calculator, is it possible to transform $10^{2017}$ to $10^{11}$ using the two mentioned keys?
Solution

a) We want the number $7 \cdot 10^{2} \cdot 10^{7}=7 \cdot 10^{9}$ to appear on the screen. One way to perform this conversion, starting with $7 \cdot 10^{2}$, is to press the key with the operation of multiplying by $10^{8}$ four times and the key for division by $10^{5}$ five times:

$7 \cdot 10^{2} \rightarrow 7 \cdot 10^{10} \rightarrow 7 \cdot 10^{18} \rightarrow 7 \cdot 10^{26} \rightarrow 7 \cdot 10^{34} \rightarrow 7 \cdot 10^{29} \rightarrow 7 \cdot 10^{24} \rightarrow 7 \cdot 10^{19} \rightarrow 7 \cdot 10^{14} \rightarrow 7 \cdot 10^{9}$.

b) We want the number $10^{10} \cdot 10^{-4} \cdot 10^{7}=10^{13}$ to appear on the screen, which is the conversion of the given number into Zimimimetros. One way to perform this conversion, starting with 1000, is to press the key with the operation of multiplying by $10^{8}$ five times and the key for division by $10^{5}$ six times:

$$
1000 \rightarrow 1000 \cdot 10^{5 \cdot 8} \rightarrow 1000 \cdot 10^{5 \cdot 8} \cdot 10^{-6 \cdot 5}=10^{3+5 \cdot 8-6 \cdot 5}=10^{13}
$$

c) If we use the key for division by $10^{5}$ five times and then use the key for multiplication by $10^{8}$ three times, we will alter the number on the screen by a factor of $10^{-5 \cdot 5+3 \cdot 8}=10^{-1}$. Therefore, repeating this exact sequence of operations $2017-11=2006$ times, we will transform the given number into:

$$
10^{2017} \rightarrow 10^{2017} \cdot 10^{-1}=10^{2016} \rightarrow \ldots \rightarrow 10^{12} \cdot 10^{-1}=10^{11}
$$