Question

How do you express `90000/0.03` in scientific notation and then divide using the rules of exponents?

Answer 1

`3*10^6`

Explanation:

In scientific notation, you have `90000 = 9*10^4` and `0.03=3*10^(-2)`. The fraction becomes

`\frac{9*10^4}{3*10^(-2)}`

We can simplify the numeric and exponential parts separately, so we can begin with

`\frac{cancel(9)^3*10^4}{cancel(3)*10^(-2)} = \frac{3*10^4}{10^(-2)}`

As for the exponential part, we use the definitions

`a^b*a^c = a^(b+c)`

`a^b/a^c = a^(b-c)`

to conclude that

`\frac{3*10^4}{10^(-2)} = 3*10^{4-(-2)} = 3*10^6`

Answer 2

`3.0xx10^6`

Explanation:

Given: `90000/0.03`

`color(brown)(" Both methods")`

`color(blue)("Without the rules of exponents for comparison")`

Lets get rid of the decimal.

`90000/0.03xx1 color(white)("d")->color(white)("d")90000/0.03xx100/100 =9000000/3`

Giving: `3000000 =3.0xx10^6`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Using rules of exponents")`

`color(purple)(90000/0.03 color(white)("d") = color(white)("d")(9xx10^4)/(3xx10^(-2))color(white)("d") =color(white)("d") 9/3xx(10^4)/(10^(-2)))`

but `1/(10^("negative 2")) = 10^("positive 2")` giving:

`color(purple)(9/3xx10^(4+2)=3.0xx10^6)`