# How to easily divide numbers in scientific notation

To solve for wavelength, I use this equation, with some values filled in (Excuse the lack of formatting, I am not aware on how to do this)

$$w = \frac{3.0 \times 10^8 m/s}{6.4 \times 10^{14} 1/s}$$

Without using a calculator, how can I easily divide $3.0 \times 10^8$ and $6.4 \times 10^{14}$? I have forgotten if $3.0 \times 10^8$ means $3$ followed by $8$ zeros, or $8$ in addition to the $3.0$? Or $3$ followed by $8$ zeros?

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$3.0 \times 10^n$ will be a $3$ followed by $n$ zeros. – anonymous Jan 12 '13 at 16:32
$3.0 \times 10^1 = 3.0 \times 10 = 30$ – Muphrid Jan 12 '13 at 16:34
And 6.4 x 10^14? Is that 64 followed by 14 zeros. Perhaps 13? – Josiah Jan 12 '13 at 16:35
$\times10^n$ means move the decimal point $n$ places to the right. – Rahul Jan 12 '13 at 18:30
@RahulNarain, Thanks a lot. This refreshed my memory. – Josiah Jan 12 '13 at 19:37

$3.0 * 10^8$ means $3$ with $8$ zeros. You can divide $3.0 / 6.4 \approx 0.47$ and subtract $8 - 14$ to get $10^{-6}$.

$0.47 * 10^{-6} = 4.7 * 10^{-7}$

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Thanks. That helps a lot. – Josiah Jan 12 '13 at 16:36
You might want to slightly alter your answer though. The division rounds to 0.47 * 10^-6. Since this is concerning science, it is quite important. :) – Josiah Jan 12 '13 at 16:41
Yes, but scientific notation is almost always used for approximations. How do you write that sign in latex? – Bojan Serafimov Jan 12 '13 at 16:43
Exactly, but then you need to round to approximate accurately. At least I'm suppose to. Maybe the "technical way" is different. You can keep it how it is, I'm just saying that the answer was 0.47 * 10^-6. Actually it was 4.7 * 10^-7 – Josiah Jan 12 '13 at 16:49

$3.0\times 10^8$ is 3 with 8 zeroes after it. Easy way to remember: figure out what $3.0\times 10^1$ would be (30), and how many zeroes it has. Extrapolate.

Regarding the division, first divide the numbers, then the exponents:

$$w = \frac{3.0 \times 10^8 m/s}{6.4 \times 10^{14} 1/s}$$

$$= \frac{(3\div6.4) \times 10^8 m/s}{10^{14} 1/s}$$ $$= \frac{0.46875 \times 10^8 m}{10^{14}}$$ $$= 0.46875 \times \frac{ 10^8 }{10^{14}} m$$

Now, rememver that when you multiply/divide exponentials with the same base, the powers are added/subtracted respectively. So, $10^x\div10^y=10^{x-y}$

$$\therefore w = 0.46875 \times 10^{-6} m =4.6875\times 10^{-7} m$$

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