Confused about order of operations to simplify $\frac15\div\frac15\div\frac15\div\frac15\div\frac15\div5\div5\div5$ (and others) Here is the question that confused me:
$$\text{What is the value of}\;\frac15\div\frac15\div\frac15\div\frac15\div\frac15\div5\div5\div5\;? \tag1$$
If the signs stay the same, is the operation done LTR or RTL? I always assumed that it would be LTR. However, divide reverses numerator and denominator so how do we handle a complex chain of operations like the above example?
What about the following operations?
$$5\div5\div5\div5\div5\div5\times5\div5\div5\div5\div5\div5 \tag2$$
and
$$5\div5\div5\div5-5\div5\div5\div5\div5\times5\div5\div5+5\div5\div5 \tag3$$
Here is the solution in the book:


As far as I know, when dealing with consecutive multiplication and division, one can simply go from left to right. This means that your example problem is solved as follows:
$$
\begin{array}
 \div \frac{1}{5} \div \frac{1}{5} \div \frac{1}{5} \div \frac{1}{5} \div \frac{1}{5}\div 5 \div 5 \div 5 &= 1 \div \frac{1}{5} \div \frac{1}{5} \div \frac{1}{5}\div 5 \div 5 \div 5 \\
&= 5 \div \frac{1}{5} \div \frac{1}{5}\div 5 \div 5 \div 5 \\
&= 25 \div \frac{1}{5}\div 5 \div 5 \div 5 \\
&= 125\div 5 \div 5 \div 5 \\
&= 25 \div 5 \div 5 \\
&= 5 \div 5 \\
&= 1 \\
\end{array}
$$
If you wanted to, you could turn the division into multiplication as you suggested:
$$
\begin{array}
 \div \frac{1}{5} \div \frac{1}{5} \div \frac{1}{5} \div \frac{1}{5} \div \frac{1}{5}\div 5 \div 5 \div 5 &= \frac{1}{5} \times 5 \times 5 \times 5 \times 5 \times \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} \\
&= \frac{5 \times 5 \times 5 \times 5}{5 \times 5 \times 5 \times 5} \\
&= 1
\end{array}
$$
