# How to Convert Scientific Notation to Standard Notation

Now we will take a closer look at how to convert from scientific notation back to standard notations. The main idea is to rewrite the power of 10, and then multiply the number $a$ by this number.

Rule

### ScientificNotationtoStandardNotation

• Calculate the power of 10 (ex. $1{0}^{4}=10\phantom{\rule{0.17em}{0ex}}000$).

• Multiply the two numbers together.

Rule

### MovingComma–ScientificNotationtoStandardNotation

• For large numbers, when $n>1$ in $1{0}^{n}$, you need to move the comma to the right to write it as a standard notation. The number will then be greater than $1$.

• For small numbers, when $n<0$ in $1{0}^{n}$, you need to move the comma to the left to write it as a standard notation. The number is then between $0$ and $1$.

Here are some examples where the power of 10 has a positive exponent. Keep in mind that the exponent is telling us how many decimal places to move the comma to the right. Note that it is the opposite way of what you did when you went from standard notation to scientific notation.

Example 1

Write $\text{}1.253\text{}\cdot 1{0}^{9}$ in standard notation.

$\begin{array}{llll}\hfill 1.253\cdot 1{0}^{9}& =1.253\cdot 1\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =1\phantom{\rule{0.17em}{0ex}}253\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

 $1.253\cdot 1{0}^{9}=1.253\cdot 1\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000=1\phantom{\rule{0.17em}{0ex}}253\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000$

Example 2

Write $\text{}6.24\text{}\cdot 1{0}^{5}$ in standard notation.

$\begin{array}{llll}\hfill 6.24\cdot 1{0}^{5}& =6.24\cdot 100\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =624\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

 $6.24\cdot 1{0}^{5}=6.24\cdot 100\phantom{\rule{0.17em}{0ex}}000=624\phantom{\rule{0.17em}{0ex}}000$

In the examples below, you can see how to proceed when a number in scientific notation has a negative exponent. You can imagine that the exponent tells us how many zeros there should be in total, or how many places you need to move the comma to the left. Note that it is the opposite of what you did when you went from standard notation to scientific notation.

Example 3

Write $\text{}6.24\text{}\cdot 1{0}^{-5}$ as a standard notation.

$\begin{array}{llll}\hfill 6.24\cdot 1{0}^{-5}& =6.24\cdot 0.000\phantom{\rule{0.17em}{0ex}}01\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =0.000\phantom{\rule{0.17em}{0ex}}062\phantom{\rule{0.17em}{0ex}}4\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

 $6.24\cdot 1{0}^{-5}=6.24\cdot 0.000\phantom{\rule{0.17em}{0ex}}01=0.000\phantom{\rule{0.17em}{0ex}}062\phantom{\rule{0.17em}{0ex}}4$

Example 4

Write $\text{}1.982\text{}\cdot 1{0}^{-3}$ in standard notation.

$\begin{array}{llll}\hfill 1.982\cdot 1{0}^{-3}& =1.982\cdot 0.001\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =0.001\phantom{\rule{0.17em}{0ex}}982\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

 $1.982\cdot 1{0}^{-3}=1.982\cdot 0.001=0.001\phantom{\rule{0.17em}{0ex}}982$

Example 5

Write $\text{}9.003\text{}\cdot 1{0}^{-6}$ in standard notation.

$\begin{array}{llll}\hfill 9.003\cdot 1{0}^{-6}& =9.003\cdot 0.000\phantom{\rule{0.17em}{0ex}}001\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =0.000\phantom{\rule{0.17em}{0ex}}009\phantom{\rule{0.17em}{0ex}}003\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

 $9.003\cdot 1{0}^{-6}=9.003\cdot 0.000\phantom{\rule{0.17em}{0ex}}001=0.000\phantom{\rule{0.17em}{0ex}}009\phantom{\rule{0.17em}{0ex}}003$