Foundational Math Skills
5 Scientific Notation
Lesson
Learning Outcomes
By the end of this chapter, learners will be able to
- explain the use of exponents,
- describe the system of scientific notation, and
- convert numbers from scientific form to standard form.
Exponents
You will see exponents being used in various ways related to health care topics. For instance, it might be for very large, or very small, amounts of medication or diagnostic test values. You may also come across exponents when reading statistical information in journal articles.
Exponents are numbers written in superscript, to the right of a number, called the base. Exponents can be positive or negative. A positive exponent is used to identify how may times the base number should be multiplied by itself. This number is referred to as the power. A negative exponent is the reciprocal of the number with a positive exponent. In general, positive exponents are related to large numbers while negative exponents are related to small numbers. While it is unlikely you will need to calculate what the power of a number equals, the following practice questions may help you to gain an appreciation for how the values of numbers change in size depending on the size of the base number and the size of the exponent.
Positive Exponent
BaseExponent
For example: [latex]2^{6}[/latex] or [latex]2\times{2}\times{2}\times{2}\times{2}\times{2}=64[/latex]
This can be read aloud as “two to the sixth power.”
Negative Exponent
[latex]\begin{align*} 2^{−6}&=\dfrac{1}{2^6} \\ \\ &= \dfrac{1}{2\times2\times2\times2\times2\times2} \\ \\ &=\dfrac{1}{64} \\ \\ &=0.015625 \end{align*}[/latex]
Sample Exercise 5.1
- Write five to the power of three.
- What does five to the power of three equal?
Answers:
- [latex]5^{3}[/latex]
- [latex]5\times{5}\times{5}=125[/latex]
Sample Exercise 5.2
What number is represented by [latex]6^{-4}[/latex]?
Answers:
[latex]\begin{align*} 6^{−4}&=\dfrac{1}{6^4} \\ \\ &=\dfrac{1}{1296} \\ \\ &=0.0007716049382716 \end{align*}[/latex]
Scientific Notation
Scientific notation is a special way of concisely expressing very large and very small numbers. You can think of it like an abbreviation of a number. When a number is not abbreviated, it is known as a number in standard form. When numbers are written in scientific notation, the base number is multiplied or divided by a power of 10 to make the number large or small. When the exponent is positive, the base number is multiplied by 10, a number of times equal to the number of the exponent. When the exponent is negative, the base number is divided by 10, a number of times equal to the number of the exponent.
Positive Exponents
number x 10n
[latex]\begin{align*} \text{a. }{3.4\times 10^{5}}&={3.4\times10\times10\times10\times10\times10} \\ \\ &=340000\end{align*}[/latex]
[latex]\begin{align*} \text{b. }{2.7\times 10^{3}}&={2.7\times10\times10\times10} \\ \\ &=2700\end{align*}[/latex]
[latex]\begin{align*} \text{c. }{7.1\times 10^{8}}&={7.1\times1