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Consider a dust mite that measures [tex]10^{-3}[/tex] millimeters in length and a gecko that measures [tex]10[/tex] centimeters long. How many orders of magnitude longer is the gecko compared to the mite?

Answer :

The gecko is [tex]\(10^2\)[/tex] times longer than the dust mite in terms of orders of magnitude.

To find out how many orders of magnitude longer the gecko is compared to the dust mite, we need to determine the difference in their lengths in scientific notation and then count the number of places the decimal point has shifted.

The length of the dust mite is [tex]\(10^{-3}\)[/tex] millimeters, and the length of the gecko is (10) centimeters, which is equivalent to [tex]\(10^2\)[/tex] millimeters.

Converting the length of the gecko to millimeters:

[tex]\[10 \, \text{centimeters} = 10^2 \, \text{millimeters} = 100 \, \text{millimeters}\][/tex]

Now, we can express both lengths in scientific notation:

Dust mite: [tex]\(10^{-3}\)[/tex] millimeters

Gecko: (100) millimeters = [tex]\(10^2\)[/tex] millimeters

The difference in their lengths in scientific notation is:

[tex]\[10^2 - 10^{-3} = 10^{2} \][/tex]

So, the gecko is [tex]\(10^2\)[/tex] times longer than the dust mite in terms of orders of magnitude.

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