Why It Matters: Exponents

Why study exponents?

Module 4: Concepts you need from module 0: 1) order of operations. 2) operations with fractions and decimals. 3) Simplify mathematical expressions. Concepts you need from module 1: problem solving.

Concepts you need from module 0, and 1.

Artist's concept of Voyager in flight.

Artist’s concept of Voyager in flight.

Mathematicians, scientists, and economists commonly encounter very large and very small numbers.  For example, Star Wars fans may remember Han Solo bragging about the Millennium Falcon’s ability to make the Kessel Run in less than 12 parsecs in Episode IV.  He was referring to a smuggler’s route with sections that were flown in hyperspace, making length an important factor in how quickly a ship could make the run.

In reality, a parsec is a unit of length used to measure large distances to objects outside the solar system. A parsec is equal to about 31 trillion kilometers, or 19 trillion miles in length. Rather than writing all the zeros associated with the number 1 trillion (1,000,000,000,000) we commonly use the written words or scientific notation, which is also called exponential notation. Scientific notation uses exponents to represent the number of zeros that come before or after the important digits of a very small or large number. Using scientific notation, 19 trillion miles would be written [latex]{1.9}\times{10}^{13}[/latex] miles. In this example the number 13 is the exponent and the number 10 is referred to as the base.

The most distant space probe, Voyager 1, was 0.0006 parsecs from Earth as of March 2015. It took Voyager 37 years to cover that distance. Voyager 1 was launched by NASA on September 5, 1977. As of 2013, the probe was moving with a relative velocity to the sun of about 17030 m/s. With the velocity the probe is currently maintaining, Voyager 1 is traveling about 325 million miles per year, or 520 million kilometers per year. Here are some more distances to well-known astronomical objects in parsecs:

  • The distance to the open cluster Pleiades is 130 parsecs from Earth. That’s [latex]{1.7}\times{10}^{15}[/latex] miles.
  • The center of the Milky Way is more than 8 kiloparsecs (a kiloparsec is 1000 parsecs) from Earth, and the Milky Way is roughly 34 kiloparsecs across.
  • The nearest star to Earth (other than the sun), Proxima Centauri, is about 1.3 parsecs from the sun.
  • Most of the stars visible to the unaided eye in the nighttime sky are within 500 parsecs of the sun.

In this section, you will learn the rules for algebraic operations in terms with exponents, then apply them to calculations involving very large or small numbers.