The weight of an ant averages about $3 \times 10^{-6} \text{ kg}$. There are about $1 \times 10^{16}$ ants in the world. What is the approximate weight in kilograms of all the ants in the world? Write the answer in scientific notation.
We are given that the average weight of a single ant is \(w_a = 3 \times 10^{-6}\) kg and the total number of ants in the world is estimated to be \(n_a = 1 \times 10^{16}\). Our objective is to find the total approximate weight \(W\) of all the ants in the world in kilograms, expressed in scientific notation. To find the total weight, we need to multiply the average weight of one ant by the total number of ants: \(W = w_a \times n_a\).
Substitute the given values into the expression for total weight:
\[W = (3 \times 10^{-6}) \times (1 \times 10^{16})\]
To calculate this, we multiply the coefficients (3 and 1) and the powers of 10 separately:
\[W = (3 \times 1) \times (10^{-6} \times 10^{16})\]
Using the laws of exponents, \(10^m \times 10^n = 10^{m+n}\), we combine the powers of 10:
\[10^{-6} \times 10^{16} = 10^{-6+16} = 10^{10}\]
Thus, the total weight is:
\[W = 3 \times 10^{10} \text{ kg}\]
Understanding how to use scientific notation for very large or small numbers is essential in fields like biology, astronomy, and physics. For example, if you know the average mass of a single cell in your body and the estimated number of cells, you can calculate your total body mass. This method allows scientists to perform complex calculations with massive quantities, such as the total mass of all humans on Earth or the total number of atoms in a galaxy, efficiently and without dealing with strings of zeros.