Question

metric measurement consider a dust mite that measures $10^{-3}$ millimeters in length and a gecko that measures 10 centimeters long. how many orders of magnitude as long as the mite is the gecko?

Explanation:

Step1: Convert units to be consistent

First, convert the gecko's length from centimeters to millimeters. Since 1 centimeter = 10 millimeters, a 10 - centimeter gecko is \(10\times10 = 10^{2}\) millimeters. The dust mite's length is \(10^{- 3}\) millimeters.

Step2: Find the ratio of lengths

To find how many orders of magnitude the gecko is longer than the mite, we can find the ratio of their lengths and then express it as a power of 10. The ratio \(r=\frac{\text{Length of gecko}}{\text{Length of mite}}=\frac{10^{2}}{10^{-3}}\). Using the rule of exponents \(\frac{a^{m}}{a^{n}}=a^{m - n}\), we have \(r = 10^{2-(-3)}=10^{5}\). The exponent of 10 in the ratio gives the number of orders of magnitude.

Answer:

5