Question

How do you simplify and write `0.0007 xx 190` in scientific notation?

Answer 1

See a solution process below:

Explanation:

First, write each term in scientific notation:

For `0.0007` we need to move the decimal point 4 places to the right therefore the exponent of the 10s term will be negative:

`0.0007 = 7.0 xx 10^-4`

For `190` we need to move the decimal point 2 places to the left therefore the exponent of the 10s term will be positive:

`190 = 1.9 xx 10^2`

We can now rewrite this expression as:

`0.0007 xx 190 => (7.0 xx 10^-4)(1.9 xx 10^2) =>`

`(7.0 xx 1.9)(10^-4 xx 10^2) => 13.3(10^-4 xx 10^2)`

We can now use this rule of exponents to combine the 10s terms:

`x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) + color(blue)(b))`

`13.3(10^color(red)(-4) xx 10^color(blue)(2)) => 13.3 xx 10^(color(red)(-4) + color(blue)(2)) => 13.3 xx 10^-2`

To put this in true scientific notation we need to move the decimal point one place to the left so we need to add `1` to the 10s exponent:

`13.3 xx 10^-2 => 1.33 xx 10^(-2 + 1) => 1.33 xx 10^-1`

Answer 2

A lot of detail provided to help with understanding.

`"1.33xx10^(-1)`

Explanation:

Note that `10^0=1" and that "10^1=10`

`0.0007->0.0007xx1/10^0`

`0.0007->0.007xx1/10^1`

`0.0007->0.07xx1/10^2`

`0.0007->0.7xx1/10^3`

`0.0007->7.0xx1/10^4`
..............................................................................

`190->190xx10^0`

`190->19.0xx10^1`
...........................................................................

So we can write: `" "0.0007xx190` as

`7xx19xx1/10^4xx10`

Not that `7xx19` is the same as `(7xx20)-7=140-7=133` giving:

`133xx1/10^4xx10" "->133xx1/10^3`

but `133" is the same as "1.33xx10^2` giving:

`1.33xx10^2/10^3`

`1.33xx1/10^" "->" "1.33xx10^(-1)`