Question

The mass of a killer whale is 950,000,000 milligrams. What is its mass, in kilograms, expressed in scientific notation?

Answer 1

`950,000,000 mg = 9.5 xx 10^2kg`

Explanation:

We can use conversion factors to get this answer. Let's begin by putting our number into scientific notation in the original units.

`950,000,000 mg = 9.5 xx 10^8 mg`

A conversion factor is a fraction that is equal to `1` such that multiplying anything by it will leave the actual quantity the same, but with different units. In our case we are trying to convert from `mg` to `kg`. We know how many `mg` are in a `g`

`1g = 1000mg implies (1g)/(1000mg) = 1 implies (1000mg)/(1g)=1`

we can also re-write these in scientific notation

`1g = 1xx10^3mg implies (1g)/(1xx10^3mg) = 1 implies (1xx10^3mg)/(1g)=1`

We can do the same for `kg` to `g`

`1kg = 1xx10^3g implies (1kg)/(1xx10^3g) = 1 implies (1xx10^3g)/(1kg)=1`

Now we just choose the fractions that have the units in the right location to cancel the units we don't want, leaving behind the units we do want:

`9.5 xx 10^8 color(red)(cancel(mg)) * (1 color(blue)(cancel(g)) )/(1xx10^3 color(red)(cancel(mg)) )*(1kg)/(1xx10^3 color(blue)(cancel(g) )`

then, multiplying powers leads us to add the exponents

`=(9.5 xx 10^8)/(1xx10^(3+3)) kg = (9.5 xx 10^8)/(1xx10^6) kg`

and and exponent in the demonimator is the same as the negative exponent in the numerator, i.e.

`=9.5 xx 10^8*1xx10^-6 kg = 9.5 xx 10^(8-6) kg`

finally

`950,000,000 mg = 9.5 xx 10^2kg`