Question

How do you simplify `\frac { ( 7.2\times 10^ { - 5} ) ( 9.6\times 10^ { 4} ) } { ( 3.6\times 10^ { - 8} ) }`?

Answer 1

See a solution process below:

Explanation:

First, rewrite the expression as:

`((7.2 xx 9.6)/3.6)((10^-5 xx 10^4)/10^-8) =>`

`((color(red)(cancel(color(black)(7.2)))2 xx 9.6)/color(red)(cancel(color(black)(3.6))))((10^-5 xx 10^4)/10^-8) =>`

`(2 xx 9.6)((10^-5 xx 10^4)/10^-8) =>`

`19.2 xx ((10^-5 xx 10^4)/10^-8)`

Next, we can use this rule of exponents to simplify the numerator of the 10s terms:

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

`19.2 xx ((10^color(red)(-5) xx 10^color(blue)(4))/10^-8) =>`

`19.2 xx 10^(color(red)(-5)+color(blue)(4))/10^-8 =>`

`19.2 xx 10^color(red)(-1)/10^-8`

Then, we can use this rule of exponents to simplify the remaining 10s terms:

`x^color(red)(a)/x^color(blue)(b) = x^(color(red)(a)-color(blue)(b))`

`19.2 xx 10^color(red)(-1)/10^color(blue)(-8) =>`

`19.2 xx 10^(color(red)(-1)-color(blue)(-8)) =>`

`19.2 xx 10^(color(red)(-1)+color(blue)(8)) =>`

`19.2 xx 10^7`

We can change this to scientific notation by moving the decimal point 1 place to the left which means we need to add to the numerator for the 10s term:

`19.2 xx 10^7 => 1.92 xx 10^8`

Answer 2

`=1.92 xx10^8`

Explanation:

We can use laws of indices first so that all the indices are positive.

Recall: `x^-m= 1/x^m and 1/x^-n = x^n`

`(7.2 xx10^-5 xx 9.6 xx10^4)/(3.6 xx10^-8)`

`=(7.2 xx 9.6 xxcolor(red)(10^4 xx10^8))/(3.6 xx10^5)" "larr` now simplify

`=(cancel7.2^color(blue)(2)color(blue)( xx 9.6) xxcolor(red)(10^12))/(cancel3.6xx10^5)`

`=color(blue)(19.2) xx10^7" "larr` adjust into correct scientific notation.

`=1.92 xx10^8`