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\times10\times10\times10\times10\times10\times10\times10\times10} \\ \\ &=710000000\end{align*}[/latex]
Negative Exponents
number × 10−n
[latex]\begin{align*} \text{a. }{4.2\times 10^{−3}}&=4.2\times\dfrac{1}{10^{3}} \\ \\ &=4.2\times \dfrac{1}{10\times 10\times 10} \\ \\\ &=\dfrac{4.2}{1000} \\ \\ &=0.0042\end{align*}[/latex]
[latex]\begin{align*} \text{b. }{9.3\times10^{−5}}&=9.3\times\dfrac{1}{10^{5}} \\ \\ &=9.3\times\dfrac{1}{10\times10\times10\times10\times10} \\ \\ &=\dfrac{9.3}{10\times10\times10\times10\times10} \\ \\ &=\dfrac{9.3}{10000} \\ \\ &=0.00093\end{align*}[/latex]
Here you can see in scientific notation, the number is divided by 10 the same number of times as the number of the exponent.
When determining what the number is which is represented by scientific notation, you can easily do this just by moving the decimal place over by the number of spaces equal to the exponent. The decimal place will move to the right with positive exponents, making the number larger, and to the left for negative exponents, making the number smaller.
Positive Exponents
Move the decimal to the right to make the number larger.
In this example, the exponent, or the power, is 5. Move the decimal five places to the right. (The numbers in subscript show the number of places the decimal is moving.)
eg. [latex]3.4\times{10}^{5}[/latex] = 3.4 1 0 2 0 3 0 4 0 5 = 340000
Negative Exponents
Move the decimal to the left to make the number smaller.
In this example, the exponent, or the power, is 3. Move the decimal three places to the right.
eg. [latex]4.2\times{10}^{-3}[/latex] = 3 0 2 0 1 4 . 2 = 0.0042
Sample Exercise 5.3
Write [latex]1.7\times{10^{6}}[/latex] in standard form.
Answers:
[latex]1700000[/latex]
Move the decimal to the right six places. ([latex]1.7\times{10^{6}}[/latex] = 1.7 1 0 2 0 3 0 4 0 5 0 6= [latex]1700000[/latex])
Key Takeaways
- Exponents are helpful when writing very large and very small numbers.
- Numbers with positive exponents will be greater or equal to one.
- Numbers with negative exponents will be less than one.
- When determining the value of a number written in scientific notation, if the power of 10 is positive you move the decimal to the right.
- When determining the value of a number written in scientific notation, if the power of 10 is negative you move the decimal to the left.
Practice Set 5.1: Determining the numerical value of numbers with positive exponents
Practice Set 5.1: Determining the numerical value of numbers with positive exponents
Calculate the value of the following numbers with exponents:
- [latex]5^{4}[/latex]
- [latex]2^{7}[/latex]
- [latex]4^{2}[/latex]
- [latex]8^{3}[/latex]
- [latex]6^{6}[/latex]
- [latex]12^{4}[/latex]
- [latex]3^{15}[/latex]
- [latex]9^{5}[/latex]
- [latex]10^{3}[/latex]
- [latex]3^{8}[/latex]
Answers:
- [latex]625[/latex]
- [latex]128[/latex]
- [latex]16[/latex]
- [latex]512[/latex]
- [latex]46656[/latex]
- [latex]20736[/latex]
- [latex]14348907[/latex]
- [latex]59049[/latex]
- [latex]1000[/latex]
- [latex]6561[/latex]
Practice Set 5.2: Determining the numerical value of numbers with negative exponents
Practice Set 5.2: Determining the numerical value of numbers with negative exponents
Calculate the value of the following numbers with exponents:
- [latex]2^{-4}[/latex]
- [latex]10^{-2}[/latex]
- [latex]4^{-3}[/latex]
- [latex]7^{-4}[/latex]
- [latex]32^{-1}[/latex]
- [latex]5^{-5}[/latex]
- [latex]3^{-5}[/latex]
- [latex]7^{-3}[/latex]
- [latex]8^{-2}[/latex]
- [latex]3.3^{-4}[/latex]
Answers:
- [latex]0.0625[/latex]
- [latex]0.01[/latex]
- [latex]0.015625[/latex]
- [latex]0.00041649312786339[/latex]
- [latex]0.03125[/latex]
- [latex]0.00032[/latex]
- [latex]0.0041152263374486[/latex]
- [latex]0.0029154518950437[/latex]
- [latex]0.015625[/latex]
- [latex]0.0084322648810503[/latex]
Practice Set 5.3: Determining the value of numbers written in scientific notation
Practice Set 5.3: Determining the value of numbers written in scientific notation
Convert each number to standard form.
- [latex]2.8\times{10^{4}}[/latex]
- [latex]7.34\times{10^{-6}}[/latex]
- [latex]4.9\times{10^{7}}[/latex]
- [latex]5.2\times{10^{3}}[/latex]
- [latex]1.54\times{10^{-4}}[/latex]
- [latex]6.241\times{10^{-8}}[/latex]
- [latex]5.9\times{10^{5}}[/latex]
- [latex]3.278\times{10^{-5}}[/latex]
- [latex]4.4\times{10^{2}}[/latex]
- [latex]8.623\times{10^{6}}[/latex]
Answers:
- [latex]28000[/latex]
- [latex]0.00000734[/latex]
- [latex]49000000[/latex]
- [latex]5200[/latex]
- [latex]0.000154[/latex]
- [latex]0.00000006241[/latex]
- [latex]590000[/latex]
- [latex]0.0000327[/latex]
- [latex]440[/latex]
- [latex]8623000[/latex]