How to solve a fraction with a numerator in exponential form and a denominator in numerical form without a calculator? The question:
"Imagine unwinding (straightening out) all of the DNA from a single typical cell and laying it "end-to-end"; then the sum total length will be approximately $2$ meters. Assume the human body has $10^{14}$ cells containing DNA. How many times would the sum total length of DNA in your body wrap around the equator of the earth."
The Earth's equator is $40,075$ km
Now I got this question right by dividing the assumed total length of DNA by the distance of the equator:
$$\frac{10^{14} \cdot 2 \ m}{40,075,000 \ m} = 4,990,642$$
The answer key says the answer to the question is "about $5 * 10^6$ times around the equator". But my question is, can I solve this question with an equation that converts the distance of the equator to exponential form to arrive at the same formatted answer as the answer key? Is there a mnemonic that makes it simple to do in your head? For example, if I used the equation:
$$\frac{10^{14} \cdot 2}{10^7 \cdot 4}$$
Then solved that equation to this:
$$\frac{10^7 \cdot 2}{4}$$
From here is it possible to get $$10^6 \cdot 5$$ (the answer) without using a calculator?
 A: Yes, it is possible. For your simpler example, $\frac{2 \cdot 10^7}{4}$, rewrite $10^7 $ as $10^1 \cdot 10^6 = 10 \cdot 10^6$. Then you have $\frac{20 \cdot 10^6}{4} = 5 \cdot 10^6$.
Now back to the original question: $$\frac{2  \cdot 10^{14}}{40,075,000}$$
First, convert the denominator to standard form (scientific notation), which is $4.0075 \cdot 10^7$. Then rewrite the numerator as $20 \cdot 10^{13}$ using the same process as before.
Then you have:
$$\frac{20  \cdot 10^{13}}{4.0075 \cdot 10^7}$$
where you can now estimate the denominator as $4 \cdot 10^7$ since you will not lose any precision, except if you are using more than $3$ sig figs. Then use the laws of indices to calculate this expression (which one is it)?
A: You have to recognize that $10=2 \cdot 5$, so $\frac{10 \cdot 2}4=5$.  You can borrow a $10$ from the $10^7$ by subtracting $1$ from the exponent.
Mental arithmetic, like so many skills, rewards practice.  Depending on the calculations you want to do, it also rewards having facts memorized so they are easy.  Do you see $1001$ and immediately think $7 \cdot 11 \cdot 13?$  Or $1000(1+0.1\%)?$  For calculations like this, approximations are acceptable.  I answered an earlier question here with the types of things I have at my fingertips.
