Question

What is `(6.73*10^5) -(4.2*10^4)`?

Answer 1

`color(saddlebrown)(6.31xx10^5)`

Explanation:

The simplest way of thinking about this is to view the `10^5" and "10^4` as 'units of measurement'.

To be able to directly subtract we need to make the units of measurement the same.

Note that

`color(green)(6.73xx10^5" is the same as ")color(brown)(6.73xx10xx10^4)color(blue)( ->67.3xx10^4)`

Write as: `(67.3xx10^4)-(4.2xx10^4)`

This is the same as: `" "(67.3-4.2)xx10^4`

`67.3`
`ul(color(white)(6)4.2) larr" Subtract"`
`63.1`

but the 'units of measurement' at this stage is `10^4` giving:

`63.1xx10^4`

Writing this in scientific notation we have:

`6.31xx10^5`

Answer 2

`6.31xx10^5`

Explanation:

Working with different operations in scientific notation is very much like working with variables in algebra.

`6.73 xx color(red)(10^5) and 4.2xxcolor(blue)(10^4)` cannot be added as they are because they are not like terms.

In the same way `6.73color(red)(x^5) and 4.2color(blue)(x^4)` are unlike terms.

The difference with numbers is that the indices can be changed by moving the decimal point.
If the point is moved to the left, the index increases.
If the point is moved to the right, the index decreases.

Use the bigger index (`x^5`)

`4.2 xx1color(blue)(0^4) = 0.42 xx color(red)(10^5)" "larr` decimal point moved to the left

Now you can add or subtract because they are like terms:

`color(white)(xxxx)6.73 xxcolor(red)(10^5)`
`color(white)(xx.)ul(-0.42xxcolor(red)(10^5)`
`color(white)(xx.x.)ul(6.31xx color(red)(10^5))" "larr` the index stays the same