8 Scientific Notation

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

Base^Exponent

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}&=\frac{1}{2^6} \\ \\ &= \frac{1}{2\times2\times2\times2\times2\times2} \\ \\ &=\frac{1}{64} \\ \\ &=0.015625 \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.

Scientific Notation has a standard visual representation when written out where the first number must be a positive number that is 1 or greater and smaller than 10.

[latex]\text{(number between 1 and 10)  }\longrightarrow \text{number } \times 10^{\; -}[/latex]

Positive Exponents

[latex]\text{ number } × 10^{n}[/latex]

[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

[latex]\text{ number } × 10^{−n}[/latex]

[latex]\begin{align*} \text{a. }{4.2\times 10^{−3}}&=4.2\times\frac{1}{10^{3}} \\ \\ &=4.2\times \frac{1}{10\times 10\times 10} \\ \\\ &=\frac{4.2}{1000} \\ \\ &=0.0042\end{align*}[/latex]

[latex]\begin{align*} \text{b. }{9.3\times10^{−5}}&=9.3\times\frac{1}{10^{5}} \\ \\ &=9.3\times\frac{1}{10\times10\times10\times10\times10} \\ \\ &=\frac{9.3}{10\times10\times10\times10\times10} \\ \\ &=\frac{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.

Scientific Notation

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.)

[latex]3.4 \times{10}^{5}  = 3.4 _{1} 0 _{2} 0 _{3} 0 _{4} 0 _{5} = 340000[/latex]

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.

[latex]4.2\times{10}^{-3} =  _{3}0 _{2} 0 _{1} 4 .2  = 0.0042[/latex]

Operations with Scientific Notation

Addition & Subtraction

When adding and/or subtracting Scientific Notation, each expression MUST have the same exponents.

If needed for the sake of facilitating the calculation, the first number of the scientific notation can be expressed as smaller than 1 or equal to 10 or higher.

Once the exponents are identical, you can add the numbers and keep the exponents the same.

Example: [latex](2.4 \times 10^{3}) + (3.5 \times 10^{3}) = 5.9 \times 10^{3}[/latex]

Multiplication

When multiplying Scientific Notation, you multiply each initial number and add the exponents.

Example: [latex](1.3 \times 10^{-5}) \times (5 \times 10^{7}) = 6.5 \times 10^{2}[/latex]

Division

When dividing Scientific Notation, you divide the first (or top) initial number by the second (or bottom) initial number and subtract the exponents.

Example: [latex](8.4 \times 10^{-3}) ÷ (2 \times 10^{5}) = 4.2 \times 10^{-8}[/latex]

When submitting a final answer, ensure rules of expressing scientific notation are followed.

Chapter Credit

Adapted from Chapter 5 Scientific Notation in A Guide for Numeracy in Nursing by Julia Langham, CC BY 4.0. The chapter in A Guide to Numeracy in Nursing was adapted from Unit 11: Exponents, Roots and Scientific Notation in the book Key Concepts of Intermediate Level Math by Meizhong Wang, shared under a CC BY 4.0 license.

Operations with Scientific Notations and Case Studies are new content.