Question

Simplify `(8.33xx10^3)+(4.1xx10^5)` giving your solution in scientific notation?

Answer 1

`4.1833 xx10^5`

Explanation:

In Algebra:
`3x^4 +8x^4 = 11x^4" "larr` add like terms

`3x^4 + 5x^6 = 3x^4 + 5x^6`
These terms cannot be added because the indices are not the same so they are not like terms

In scientific notation you do exactly the same:

`3.5 xx 10^7 + 1.3xx10^7 = 4.8 xx 10^7`

But sometimes adding creates a small problem:

`4.9xx10^5 + 7.3xx10^5 = color(red)(12.2xx10^7)" "larr` must be one digit
`color(white)(xxxxxxxxxxxxx.x) = color(red)(1.22 xx 10^8)" "larr` adjusted

`(8.33 xx 10^3) + (4.1 xx 10^5)" "larr` cannot be added like this
But unlike algebra, in scientific notation, you can change the value of the index by moving the decimal point.

If it moves to the left, the index increases.
If it moves to the right, the index decreases

=`(color(blue)(8.33 xx 10^3)) + (4.1 xx color(blue)(10^5))" "larr` make indices the same

=`0.0833 xx10^5 +4.1xx 10^5" "larr` change to the bigger index

=`4.1833 xx10^5" "larr` add like terms

What you want to avoid is writing the numbers in numeral form (with lots of zero's) adding them and then changing them back to standard form. While that might work for small indices, it is simply not practical with bigger indices!

Answer 2

`4.1833xx10^5`

Explanation:

Given:`" "(8.33xx10^3)+(4.1xx10^5)`..........Equation(1)

`color(green)("Note that "10^2xx10^3 -> 10^(2+3) = 10^5)`

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

Write equation(1) as:

`(8.33xx10^3color(magenta)(xx1))+(4.1xx10^5)`

However, 1 is the same as `10^2/10^2` giving:

`(8.33xx10^3color(magenta)(xx10^2/10^2))+(4.1xx10^5)`

Swap round the numbers (condition of being 'commutative' )

`(8.33/(color(magenta)(10^2))xx10^3color(magenta)(xx10^2))+(4.1xx10^5)`

But `8.33/10^2 = 0.0833` and `10^2xx10^3=10^5` giving

`(0.0833xx10^5)+(4.1xx10^5)`

Factor out the `10^5`

`10^5(0.0833+4.1) = 4.1833xx10^5`