What if you knew that a milligram was one-millionth of a kilogram? How could you express this relationship exponentially? After completing this Concept, you'll be able to solve real-world problems like this one that involve scientific notation.

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CK-12 Foundation: 0809S Applications Using Scientific Notation

### Guidance

Let's look at some real-world applications involving scientific notation.

#### Example A

*The mass of a single lithium atom is approximately one percent of one millionth of one billionth of one billionth of one kilogram. Express this mass in scientific notation.*

**Solution**

We know that a *percent* is \begin{align*}\frac{1}{100}\end{align*}, and so our calculation for the mass (in kg) is:

\begin{align*}\frac{1}{100} \times \frac{1}{1,000,000} \times \frac{1}{1,000,000,000} \times \frac{1}{1,000,000,000} = 10^{-2} \times 10^{-6} \times 10^{-9} \times 10^{-9}\end{align*}

Next we use the product of powers rule we learned earlier:

\begin{align*}10^{-2} \times 10^{-6} \times 10^{-9} \times 10^{-9} = 10^{((-2) + (-6) + (-9) + (-9))} = 10^{-26} \ kg.\end{align*}

The mass of one lithium atom is approximately \begin{align*}1 \times 10^{-26} \ kg\end{align*}.

#### Example B

*You could fit about 3 million \begin{align*}E\end{align*}. coli bacteria on the head of a pin. If the size of the pin head in question is* \begin{align*}1.2 \times 10^{-5} \ m^{2}\end{align*}, *calculate the area taken up by one \begin{align*}E\end{align*}. coli bacterium. Express your answer in scientific notation*

**Solution**

Since we need our answer in scientific notation, it makes sense to convert 3 million to that format first:

\begin{align*}3,000,000 = 3 \times 10^6\end{align*}

Next we need an expression involving our unknown, the area taken up by one bacterium. Call this \begin{align*}A\end{align*}.

\begin{align*}& 3 \times 10^6 \cdot A = 1.2 \times 10^{-5} && - since \ 3 \ million \ of \ them \ make \ up \ the \ area \ of \ the \ pin-head\end{align*}

Isolate \begin{align*}A\end{align*}:

\begin{align*}& A = \frac{1}{3 \times 10^6} \cdot 1.2 \times 10^{-5} && - rearranging \ the \ terms \ gives:\\ & A = \frac{1.2}{3} \cdot \frac{1}{10^6} \times 10^{-5} && - then \ using \ the \ definition \ of \ a \ negative \ exponent:\\ & A = \frac{1.2}{3} \cdot 10^{-6} \times 10^{-5} && - evaluate \And \ combine \ exponents \ using \ the \ product \ rule:\\ & A = 0.4 \times 10^{-11} && - but \ we \ can't \ leave \ our \ answer \ like \ this, \ so \ldots\end{align*}

The area of one bacterium is \begin{align*}4.0 \times 10^{-12} \ m^{2}\end{align*}.

(Notice that we had to move the decimal point over one place to the right, subtracting 1 from the exponent on the 10.)

**Evaluate Expressions in Scientific Notation Using a Graphing Calculator**

All scientific and graphing calculators can use scientific notation, and it’s very useful to know how.

To insert a number in scientific notation, use the **[EE]** button. This is **[2nd] [,]** on some TI models.

For example, to enter \begin{align*}2.6 \times 10^5\end{align*}, enter 2.6 **[EE]** 5. When you hit **[ENTER]** the calculator displays 2.6E5 if it’s set in **Scientific** mode, or 260000 if it’s set in **Normal** mode.

(To change the mode, press the ‘Mode’ key.)

#### Example C

*Evaluate \begin{align*}(2.3 \times 10^6) \times (4.9 \times 10^{-10})\end{align*} using a graphing calculator.*

**Solution**

Enter 2.3 **[EE]** \begin{align*}6 \times 4.9\end{align*} **[EE]** - 10 and press **[ENTER]**.

The calculator displays 6.296296296E16 whether it’s in Normal mode or Scientific mode. That’s because the number is so big that even in Normal mode it won’t fit on the screen. The answer displayed instead isn’t the precisely correct answer; it’s rounded off to 10 significant figures.

Since it’s a repeating decimal, though, we can write it more efficiently *and* more precisely as \begin{align*}6. \overline{296} \times 10^{16}\end{align*}.

Watch this video for help with the Examples above.

CK-12 Foundation: Applications using Scientific Notation

### Vocabulary

- In
**scientific notation**, numbers are always written in the form \begin{align*}a \times 10^b\end{align*} , where \begin{align*}b\end{align*} is an integer and \begin{align*}a\end{align*} is between 1 and 10 (that is, it has exactly 1 nonzero digit before the decimal).

### Guided Practice

*Evaluate* \begin{align*}(4.5 \times 10^{14})^3\end{align*} *using a graphing calculator.*

**Solution**

Enter (4.5 **[EE]** \begin{align*}14)^{\land} 3\end{align*} and press **[ENTER]**.

The calculator displays 9.1125E43. The answer is \begin{align*}9.1125 \times 10^{43}\end{align*}.

### Practice

For questions 1-9, use a calculator to evaluate the expression.

- \begin{align*}(3.5 \times 10^4) \cdot (2.2 \times 10^7)\end{align*}
- \begin{align*}\frac{2.1 \times 10^9}{3 \times 10^2}\end{align*}
- \begin{align*}(3.1 \times 10^{-3}) \cdot (1.2 \times 10^{-5})\end{align*}
- \begin{align*}\frac{7.4 \times 10^{-5}}{3.7 \times 10^{-2}}\end{align*}
- \begin{align*}12,000,000 \times 400,000\end{align*}
- \begin{align*}3,000,000 \times 0.00000000022\end{align*}
- \begin{align*}\frac{17,000}{680,000,000}\end{align*}
- \begin{align*}\frac{25,000,000}{0.000000005}\end{align*}
- \begin{align*}\frac{0.0000000000042}{0.00014}\end{align*}

- The moon is approximately a sphere with radius \begin{align*}r = 1.08 \times 10^3 \ miles\end{align*}. Use the formula Surface Area \begin{align*}= 4 \pi r^2\end{align*} to determine the surface area of the moon, in square miles. Express your answer in scientific notation, rounded to two significant figures.
- The charge on one electron is approximately \begin{align*}1.60 \times 10^{19}\end{align*} coulombs. One
**Faraday**is equal to the total charge on \begin{align*}6.02 \times 10^{23}\end{align*} electrons. What, in coulombs, is the charge on one Faraday? - Proxima Centauri, the next closest star to our Sun, is approximately \begin{align*}2.5 \times 10^{13}\end{align*} miles away. If light from Proxima Centauri takes \begin{align*}3.7 \times 10^4\end{align*} hours to reach us from there, calculate the speed of light in miles per hour. Express your answer in scientific notation, rounded to 2 significant figures.