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Manipulating Exponents

A powerful language for the very large and the very small.

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In physics, chemistry, and many other sciences, measurements can be either large enough or small enough that it would be difficult to compare them without the use of exponents. Therefore, in Scientific Notation, the magnitude of a number, represented with an exponent with base 10, is separated away from a scale factor between 1 and 10. This notation makes it far easier to compare the relative sizes of two numbers.

For example, which is greater:

\[\frac{8 \times 10^{8}}{2 \times 10^{2}} \ \ \text{or} \ \ \frac{9 \times 10^{9}}{3 \times 10^{3}}?\]

For another example, here is a problem that might help you visualize how small the cells in your hand are:

A hand with a diameter of \(1 \times 10^{-1} \text{ m}\) contains a cell with a diameter of \(1 \times 10^{-5} \text{ m}\). If the hand were enlarged to be about the diameter of the earth, \(1 \times 10^7\text{ m},\) the diameter of the cell would be about how large -- as large as the diameter of a soccer ball, the length of a field, the height of the world’s tallest building, or the length of the United States?

Additionally, this chapter will dive deeply into the general exploration of exponents. In addition to negative, positive, and fractional exponents, we’ll also play around with and make sense of all of the algebraic rules for working with exponents including:

  • The Product Rule
  • The Quotient Rule
  • The Power Rule, and
  • The Tower Rule

For a quick example, do you know how to quickly simplify: \[\large \left(a^3\right)^2 \div \left(a^2\right)^3? \]

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