Scientific Notation

Section 5.3 Scientific Notation

Objective: Multiply and divide expressions using scientific notation and exponent properties.

One application of exponent properties comes from scientific notation. Scientific notation is used to represent really large or really small numbers. An example of really large numbers would be the distance that light travels in a year in miles. An example of really small numbers would be the mass of a single hydrogen atom in grams. Doing basic operations such as multiplication and division with these numbers would normally be very combersome. However, our exponent properties make this process much simpler.

First we will take a look at what scientific notation is. Scientific notation has two parts, a number between one and ten (it can be equal to one, but not ten), and that number multiplied by ten to some exponent.

\begin{equation*} \textbf{ Scientific Notation: } \mathbf{a \times 10^b} \textbf{ where } \mathbf{ 1\leq a\lt 10} \end{equation*}

The exponent, $b\text{,}$ is very important to how we convert between scientific notation and normal numbers, or standard notation. The exponent tells us how many times we will multiply by $10\text{.}$ Multiplying by $10$ in affect moves the decimal point one place. So the exponent will tell us how many times the exponent moves between scientific notation and standard notation. To decide which direction to move the decimal (left or right) we simply need to remember that positive exponents mean in standard notation we have a big number (bigger than ten) and negative exponents mean in standard notation we have a small number (less than one).

Keeping this in mind, we can easily make conversions between standard notation and scientific notation.

Example 5.3.1.

\begin{align*} \text{Convert $14,200$ to scientific notation} \amp \quad \text{ Put decimal after first nonzero number } \\ 1.42 \amp\quad \text{ Exponent is how many times decimal moved, } 4 \\ \times 10^4 \amp\quad \text{ Positive exponent, standard notation is big } \\ 1.42 \times 10^4 \amp\quad \text{ Our Solution } \end{align*}

Example 5.3.2.

\begin{align*} \text{Convert $0.0042$ to scientific notation} \amp \quad \text{ Put decimal after first nonzero number } \\ 4.2 \amp\quad \text{ Exponent is how many times decimal moved, } 3 \\ \times 10^{-3} \amp\quad \text{ Negative exponent, standard notation is small } \\ 4.2 \times 10^{-3} \amp\quad \text{ Our Solution } \end{align*}

Example 5.3.3.

\begin{align*} \text{Convert $3.21 \times 10^5$ to standard notation} \amp \quad \text{Positive exponent means standard notation } \\ \amp\quad \text{ big number. Move decimal right $5$ places } \\ 321,000 \amp\quad \text{ Our Solution } \end{align*}

Example 5.3.4.

\begin{align*} \text{Convert $7.4 \times 10^{-3}$ to standard notation} \amp \quad \text{Negative exponent means standard notation } \\ \amp\quad \text{ is a small number. Move decimal left $3$ places } \\ 0.0074 \amp\quad \text{ Our Solution } \end{align*}

Converting between standard notation and scientific notation is important to understand how scientific notation works and what it does. Here our main interest is to be able to multiply and divide numbers in scientific notation using exponent properties. The way we do this is first do the operation with the front number (multiply or divide) then use exponent properties to simplify the $10$’s. Scientific notation is the only time where it will be allowed to have negative exponents in our final solution. The negative exponent simply informs us that we are dealing with small numbers. Consider the following examples.

Example 5.3.5.

\begin{align*} \left(2.1\times 10^{-7}\right)\left(3.7 \times 10^5\right) \amp \quad \text{ Deal with numbers and $10$'s separately } \\ (2.1)(3.7) = 7.77 \amp\quad \text{ Multiply numbers } \\ 10^{-7}10^5 = 10^{-2} \amp\quad \text{ Use product rule on $10$'s and add exponents } \\ 7.77 \times 10^{-2}\amp\quad \text{ Our Solution } \end{align*}

Example 5.3.6.

\begin{align*} \frac{4.96 \times 10^4}{3.1 \times 10^{-3}} \amp \quad \text{ Deal with numbers and $10$'s separately } \\ \frac{4.96}{3.1 }=1.6 \amp\quad \text{ Divide Numbers } \\ \frac{10^4}{10^{-3}}=10^7 \amp\quad \text{ Use quotient rule to subtract exponents, be careful with negatives! } \\ \amp\quad \text{ Be careful with negatives, } 4-(-3)=4+3=7 \\ 1.6 \times 10^7 \amp\quad \text{ Our Solution } \end{align*}

Example 5.3.7.

\begin{align*} \left(1.8 \times 10^{-4}\right)^3 \amp \quad \text{ Use power rule to deal with numbers and $10$'s separately } \\ 1.8^3 = 5.832 \amp\quad \text{ Evaluate } 1.8^3 \\ \left(10^{-4}\right)^3 = 10^{-12} \amp\quad \text{ Multiply exponents } \\ 5.832 \times 10^{-12} \amp\quad \text{ Our Solution } \end{align*}

Often when we multiply or divide in scientific notation the end result is not in sci- entific notation. We will then have to convert the front number into scientific notation and then combine the $10$’s using the product property of exponents and adding the exponents. This is shown in the following examples.

Example 5.3.8.

\begin{align*} \left (4.7 \times 10^{-3}\right)\left(6.1 \times 10^9\right) \amp \quad \text{ Deal with numbers and $10$'s separately } \\ (4.7)(6.1) = 28.67 \amp\quad \text{ Multiply numbers } \\ 2.867 \times 10^1 \amp\quad \text{ Convert this number into scientific notation } \\ 10^1 10^{-3} 10^9 = 10^7 \amp\quad \text{ Use product rule, add exponents, using $10^1$ from conversion } \\ 2.867 \times 10^7\amp\quad \text{ Our Solution } \end{align*}

World View Note: Archimedes ($287$ BC - $212$ BC), the Greek mathematician, developed a system for representing large numbers using a system very similar to scientific notation. He used his system to calculate the number of grains of sand it would take to fill the universe. His conclusion was $10^{63}$ grains of sand because he figured the universe to have a diameter of $10^{14}$ stadia or about $2$ light years.

Example 5.3.9.

\begin{align*} \frac{2.014 \times 10^{-3}}{3.8 \times 10^{-7}} \amp \quad \text{ Deal with numbers and $10$'s separately } \\ \frac{2.014 }{3.8}=0.53 \amp\quad \text{ Divide numbers } \\ 0.53 = 5.3 \times 10^{-1} \amp\quad \text{ Change this number into scientific notation } \\ \frac{10^{-1}10^{-3}}{10^{-7} } =10^3 \amp\quad \text{ Use product and quotient rule, using $10^{-1}$ from the conversion } \\ \amp\quad \text{ Be careful with signs: } \\ \amp\quad \text{ (-1)+(-3)-(-7)=(-1)+(-3)+7=3 } \\ 5.3 \times 10^3 \amp\quad \text{ Our Solution } \end{align*}

Exercises Exercises - Integers

Exercise Group.

Write each number in scientific notiation.

1.

$885 $

2.

$0.000744$

3.

$0.081$

4.

$1.09$

5.

$0.039 $

6.

$15000$

Exercise Group.

Write each number in standard notation.

7.

$8.7 \times 10^5$

8.

$2.56 \times 10^2$

9.

$9 \times 10^{-4}$

10.

$5 \times 10^4$

11.

$2 \times 10^0$

12.

$6 \times 10^{-5}$

Exercise Group.

Simplify. Write each answer in scientific notation.

13.

$\left(7 \times 10^{-1}\right)\left(2 \times 10^{-3}\right)$

14.

$\left(2 \times 10^{-6}\right)\left(8.8 \times 10^{-5}\right)$

15.

$\left(5.26 \times 10^{-5}\right)\left(3.16 \times 10^{-2}\right) $

16.

$\left(5.1 \times 10^6\right)\left(9.84 \times 10^{-1}\right) $

17.

$\left(2.6 \times 10^{-2}\right)\left(6 \times 10^{-2}\right) $

18.

$\frac{7.4 \times 10^4}{1.7\times 10^{-4}} $

19.

$\frac{4.9 \times 10^1 }{2.7\times 10^{-3} }$

20.

$\frac{7.2\times 10^{-1} }{7.32\times 10^{-1} }$

21.

$\frac{5.33\times 10^{-6}}{9.62\times 10^{-2}} $

22.

$\frac{ 3.2\times 10^{-3}}{5.02 \times 10^0 } $

23.

$\left(5.5 \times 10^{-5}\right)^2$

24.

$\left(9.6 \times 10^3\right)^{-4} $

25.

$\left(7.8 \times 10^{-2}\right)^5 $

26.

$\left(5.4 \times 10^6\right)^{-3}$

27.

$\left(8.03 \times 10^4\right)^{-4} $

28.

$\left(6.88 \times 10^{-4}\right)\left(4.23 \times 10^1\right)$

29.

$\frac{6.1\times 10^{-6}}{5.1\times 10^{-4} } $

30.

$\frac{8.4\times 10^5 }{ 7\times 10^{-2}} $

31.

$\left(3.6 \times 10^0\right)\left(6.1 \times 10^{-3}\right) $

32.

$\left(3.15 \times 10^3\right)\left(8 \times 10^{-1}\right) $

33.

$\left(1.8 \times 10^{-5}\right)^{-3}$

34.

$\frac{9.58\times 10^{-2} }{1.14\times 10^{-3} }$

35.

$\frac{9\times 10^4 }{7.83\times 10^{-2} } $

36.

$\left(8.3 \times 10^1\right)^5 $

37.

$\frac{3.22\times 10^{-3} }{7\times 10^{-6} } $

38.

$\frac{5\times 10^6 }{6.69 \times 10^2 } $

39.

$\frac{2.4\times 10^{-6} }{ 6.5 \times 10^0} $

40.

$\left(9 \times 10^{-2}\right)^{-3} $

41.

$\frac{6\times 10^3 }{ 5.8\times 10^{-3}} $

42.

$\left(2\times 10^4 \right)\left(6\times 10^1\right) $