Question

Write these numbers in scientific notation: (a) the U.S. population, \(290000000 ;\) (b) the diameter of a helium nucleus, $0.0000000000000038 \mathrm{m}$

Short answer

Question: Convert the following numbers into scientific notation: (a) The U.S. population: \(290000000\) (b) The diameter of a helium nucleus: \(0.0000000000000038 \mathrm{m}\) Answer: (a) U.S. population: \(2.9 \times 10^{8}\) (b) Diameter of a helium nucleus: \(3.8 \times 10^{-14} \mathrm{m}\)

Step-by-step solution

Identify the numeric values

First, let's identify the given numeric values for both the U.S. population and the diameter of a helium nucleus. (a) The U.S. population: \(290000000\) (b) The diameter of a helium nucleus: \(0.0000000000000038 \mathrm{m}\)

Rewrite the numbers in the form of a times 10 raised to the power of b

Now we need to convert each of these numbers into the form of \(a \times 10^{b}\). We will move the decimal point to the left or right to create a number between 1 and 10 and then multiply it by 10 raised to the power representing how many times we moved the decimal point. (a) For the U.S. population: 1. Move the decimal point 8 places to the left: \(2.9\) 2. Multiply the new number by \(10^{8}\): \(2.9 \times 10^{8}\) (b) For the diameter of a helium nucleus: 1. Move the decimal point 14 places to the right: \(3.8\) 2. Multiply the new number by \(10^{-14}\): \(3.8 \times 10^{-14} \mathrm{m}\)

Write the final answers in scientific notation

Now we will write both numbers in their final scientific notation form: (a) U.S. population: \(2.9 \times 10^{8}\) (b) Diameter of a helium nucleus: \(3.8 \times 10^{-14} \mathrm{m}\)