বৈজ্ঞানিক সংখ্যাপদ্ধতি: ভাগ

(Video) Dividing in Scientific Notation

by HCCMathHelp

This video by HCC Math Help covers dividing in scientific notation.

$\large{\text{Summary}}$

The concept is the same as multiplying numbers in scientific notation; separate the two parts and deal with them individually.

Let's see this in action with an example. $$\frac{3.6\times10^5}{1.2\times10^{15}}$$ We want to divide the constant parts and divide the power of ten parts. Thus we break this into the product of two fractions.

$$\begin{align} &=\frac{3.6}{1.2}\times\frac{10^5}{10^{15}}\\ &=3\times10^{5-15}\\ &=3\times10^{-10} \end{align}$$

To get from $\frac{10^5}{10^{15}}$ to $10^{-10}$ we need to use the quotient rule for exponents. This is what allows us to subtract the powers and keep the base $10$.

(Video) Scientific Notation Division

by mathman1024

Here's a video by mathman1024 that shows you how to divide numbers in scientific notation.

$\large{\text{Summary}}$

Here is the general formula for dividing: $$\frac{a\times10^m}{b\times10^n}=\frac{a}b\times10^{m-n}\\ \text{where }1\leq \frac{a}b<10$$

The catch is, sometimes we have to move the decimal point at the end. That ensures our final answer is still in scientific notation.

Example

Here is an example from the video.

$$\begin{align} &\frac{6\times10^{12}}{1.5\times10^3}\\ &=\frac{6}{1.5}\times\frac{10^{12}}{10^3}\\ &=4\times10^{12-3}\\ &=4\times10^9 \end{align}$$

The number $4$ in front is between $1$ and $10$. We don't have to move the decimal point at all.

Another Example

Here's an example with decimal numbers not in the form we want. The problem is, $\dfrac{3\times10^7}{5\times10^{-21}}$. $$\begin{align} &\frac{3\times10^7}{5\times10^{-21}}\\ &=\frac35\times\frac{10^7}{10^{-21}}\\ &=0.6\times10^{7-(-21)}\\ &=0.6\times10^{28} \end{align}$$ But this isn't correct yet because $0.6$ is not between $1$ and $10$. So we have to move the decimal to the right by one place. $$0.6\rightarrow6$$

Since we made our wallet bigger we had to take money out of the bank so therefore we decrease our exponent. So the correct answer is: $$6\times10^{27}$$

Let's take a look at a division problem. For these problems, our numerator and denominator will be in scientific notation. Then we want our final answer to be in scientific notation as well.

$$\frac{(3\times10^2)}{(6\times10^5)}$$

Like multiplication, we can divide the constants and then divide the powers of $10$. $$\left(\frac36\right)\times\left(\frac{10^2}{10^5}\right)$$

When dividing powers with the same base, you can just subtract the exponents. This is called the Quotient Rule. $$\begin{align} &\left(\frac36\right)\times\left(\frac{10^2}{10^5}\right)\\ &=0.5\times\left(\frac{10^2}{10^5}\right)\\ &=0.5\times10^{2-5}\\ &=0.5\times10^{-3} \end{align}$$ But be careful, $0.5$ isn't between $1$ and $10$. We have to move the decimal place to the right one space. $$\begin{align} &0.5\times10^{-3}\\ &=5\times10^{-1}\times10^{-3}\\ &=5\times10^{-1+-3}\\ &=\boxed{5\times10^{-4}} \end{align}$$