A hydrogen atom has a radius of about \(5 \times 10^{-9} \mathrm{~cm}\). The radius of the observable universe is about 14 billion light-years. How many times larger than a hydrogen atom is the observable universe? Use powers-of- ten notation.

Scientific notation is a way of expressing very large or very small numbers that would be cumbersome to write out fully. By using this method, numbers are written as a product of a number between 1 and 10 and a power of ten. For example, the number 500 can be written as 5 times 10 squared, or in scientific notation as \(5 \times 10^{2}\).

When dealing with sizes as different as a hydrogen atom and the universe, scientific notation becomes indispensable. It allows us to express the radius of a hydrogen atom as \(5 \times 10^{-9}\) cm, where the exponent shows how many places to move the decimal point. In positive exponents, we move to the right for large numbers, and for negative exponents, we move to the left to represent very small numbers. This is critical when calculating the relative size difference and helps us maintain accuracy without getting lost in a sea of zeros.

Understanding the concept of scientific notation is crucial in various fields of science and helps us comprehend and compare figures that span the vast range of scales in our universe.

The conversion of light years to centimeters is a process that involves understanding what a light year measures and knowing the speed of light. A light year is a distance that light, which travels at an immense speed, covers in one year.

To convert light years to a more standard unit like centimeters, we take the speed of light in centimeters per second \(3.0 \times 10^{10}\) cm/s and multiply it by the number of seconds in a year (approximately \(3 \times 10^{7}\) seconds). Simplifying this, we find that one light year is \(9 \times 10^{17}\) cm, which is an essential factor in our calculations.

Using this conversion factor becomes necessary when we compare astronomical objects with very human-scale objects. This step bridges the gap between our everyday experience and the incomprehensible scales of the cosmos, allowing us to place things in a perspective that, although vast, is quantifiable.

The astronomical scale involves sizes and distances that can be difficult to grasp because they are so vastly different from our everyday experiences. For instance, the radius of the observable universe can be about 14 billion light years, a figure so enormous that it defies easy comprehension.

The size of a hydrogen atom, on the other hand, is incredibly small, indicated by its radius of approximately \(5 \times 10^{-9}\) cm. To contrast the universe with a hydrogen atom signifies working across scales that stretch our imagination, which is why scientific notation is so critical in expressing these figures in a manageable way.

By calculating the number of times the universe is larger than a hydrogen atom, we gain a sense of the extraordinary expanse of the cosmos. With a resulting ratio of \(2.52 \times 10^{37}\), this exercise not only enables us to compare sizes between the atom and the universe but also provides a startling visual of the incomprehensible vastness that surrounds us.