How do you write `0.0001` in scientific notation?

In scientific notation, we write a number so that it has single digit to the left of decimal sign and is multiplied by an integer power of `10`.

Note that moving decimal `p` digits to right is equivalent to multiplying by `10^p` and moving decimal `q` digits to left is equivalent to dividing by `10^q`.

Hence, we should either divide the number by `10^p` i.e. multiply by `10^(-p)` (if moving decimal to right) or multiply the number by `10^q` (if moving decimal to left).

In other words, it is written as `axx10^n`, where `1<=a<10` and `n` is an integer.

To write `0.0001` in scientific notation, we will have to move the decimal point four points to right, which literally means multiplying by `10^4`.

Hence in scientific notation `0.0001=1.0xx10^(-4)` (note that as we have moved decimal one point to right we are multiplying by `10^(-4)`.

`color(brown)("Multiply by 1 by but in the form of "1=10000/10000)`
`color(brown)("This does not change the value but it does change the way it looks.")`

`" "0.0001" "=" "0.0001xx10000/10000`

`" "=(0.0001xx10000)xx1/10000`

`" " = 1.0/10000" "=" "1.0/10^4`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`" Another way of writing "1.0/10^4" is "1.0xx10^(-4)`

Decimals are a form of writing fractions which have powers of `10` as their denominators.

`0.0001 = 1/(10,000) = 1/10^4`

Using the law of indices: `x^-1 = 1/x`, we can get rid of the fraction:

`1/10^4 = 1xx10^-4" "larr` this is scientific notation.

A short way of changing to scientific notation is to move the decimal point until there is only one (non-zero) digit to the left of the point. The number of places moved is the index.

Point moves to the right, the index decreases.
Point moves to the left, the index increases.

`0color(blue)(.000)1 = 1xx10^-4`