# 8.6: Scientific Notation with a Calculator

**Basic**Created by: CK-12

Did you know that there are about 7 billion people living on the Earth? Also, did you know that each person is made up of about 100 trillion cells? To find the total number of cells making up all the people on Earth, you would have to multiply these two numbers together, but without a calculator, that might be difficult. In this Concept, you'll learn how to evaluate expressions using scientific notation with a calculator so that you can perform multiplication problems such as this one.

### Guidance

Scientific and graphing calculators make scientific notation easier. To compute scientific notation, use the **[EE]** button. This is **[2nd] [,]** on some TI models or \begin{align*}[10^\chi]\end{align*}, which is **[2nd] [log]**.

#### Example A

*Evaluate \begin{align*}2.6 \times 10^5\end{align*} using a scientific calculator.*

**Solution:**

To enter \begin{align*}2.6 \times 10^5\end{align*}, enter 2.6 **[EE]** 5.

When you hit **[ENTER]** the calculator displays \begin{align*}2.6E5\end{align*} if it’s set in **Scientific** mode OR it displays 260,000 if it’s set in **Normal** mode.

**Evaluating Expressions Using Scientific Notation**

We can use scientific notations in expressions, equations and operations. Scientific notation makes it easier to multiply two large numbers together, or divide a decimal by another decimal. This is because when evaluating expressions with scientific notation, you can keep the powers of 10 together and deal with them separately.

#### Example B

*Simplify \begin{align*}(3.2 \times 10^6) \cdot (8.7 \times 10^{11})\end{align*}.*

**Solution:**

Start by evaluating by hand, and then check on your calculator:

\begin{align*}(3.2 \times 10^6) \cdot (8.7 \times 10^{11}) = 3.2 \times 8.7 \cdot 10^6 \times 10^{11} = 27.84 \times 10^{17}=2.784 \times 10^1 \times 10^{17} = 2.784 \times 10^{18}\end{align*}

It is necessary to keep one number before the decimal point, to be in correct scientific notation. In order to do that, we had to use the fact that \begin{align*}27.84=2.784 \times 10^1\end{align*} so we could simplify the expression fully.

In this case, the power is so large that the answer shows up in scientific notation. It does not matter if the calculator is set to show numbers in scientific notation or normally.

#### Example C

*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 percent means we divide by 100, 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 in a previous Concept.

\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*}.

Check the answer on your calculator:

### Guided Practice

*Evaluate the following expressions by hand and check your answer using a scientific calculator.*

(a) \begin{align*}(1.7 \times 10^6) \cdot (2.7 \times 10^{-11})\end{align*}

(b) \begin{align*}(1.03 \times 10^8) \div (-3.25 \times 10^{-5})\end{align*}

**Solution:**

(a) \begin{align*}(1.7 \times 10^6) \cdot (2.7 \times 10^{-11})=1.7 \times 2.7 \cdot 10^6 \times 10^{-11}=4.59 \times 10^{-5}\end{align*}

(b) \begin{align*}(1.03 \times 10^8) \div (-3.25 \times 10^{-5})&=\frac{1.03 \times 10^8}{-3.25 \times 10^{-5}}=\frac{1.03}{-3.25} \times \frac{10^8}{10^{-5}}\\ = -0.317 \times 10^{8-(-5)} &=-0.317 \times 10^{13}=-3.17 \times 10^{12}\end{align*}

You must remember to keep the powers of ten together, and have 1 number before the decimal.

### Practice

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Scientific Notation (14:26)

Evaluate the following expressions by hand and check your answers using a graphing calculator.

- \begin{align*}(3.2 \times 10^6) \cdot (8.7 \times 10^{11})\end{align*}
- \begin{align*}(5.2 \times 10^{-4}) \cdot (3.8 \times 10^{-19})\end{align*}
- \begin{align*}(1.7 \times 10^6) \cdot (2.7 \times 10^{-11})\end{align*}
- \begin{align*}(3.2 \times 10^6) \div (8.7 \times 10^{11})\end{align*}
- \begin{align*}(5.2 \times 10^{-4}) \div (3.8 \times 10^{-19})\end{align*}
- \begin{align*}(1.7 \times 10^6) \div (2.7 \times 10^{-11})\end{align*}
- \begin{align*}(4.7 \times 10^3) \cdot (2.35 \times 10^{-27})\end{align*}
- \begin{align*}(1.05 \times 10^{-16}) \cdot (7.003 \times 10^{21})\end{align*}
- \begin{align*}(2.09 \times 10^5) \div (5.006 \times 10^{-3})\end{align*}
- \begin{align*}(9.01 \times 10^{22}) \div (2.6 \times 10^{33})\end{align*}

**Mixed Review**

- 14 milliliters of a 40% sugar solution was mixed with 4 milliliters of pure water. What is the concentration of the mixture?
- Solve the system \begin{align*}\begin{cases} 6x+3y+18\\ -15=11y-5x \end{cases}\end{align*}.
- Graph the function by creating a table: \begin{align*}f(x)=2x^2\end{align*}. Use the following values for \begin{align*}x: -5 \le x \le 5\end{align*}.
- Simplify \begin{align*}\frac{5a^6 b^2 c^{-6}}{a^{11} b}\end{align*}. Your answer should have only positive exponents.
- Each year Americans produce about 230 million tons of trash (Source: http://www.learner.org/interactives/garbage/solidwaste.html). There are 307,006,550 people in the United States. How much trash is produced per person per year?
- The volume of a 3-dimesional box is given by the formula: \begin{align*}V=l(w)(h)\end{align*}, where \begin{align*}l=\end{align*}
*length,*\begin{align*}w=\end{align*}*width,*and \begin{align*}h=\end{align*}*height*of the box. The box holds 312 cubic inches and has a length of 12 inches and a width of 8 inches. How tall is the box?

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### Image Attributions

Here you'll learn how to use a calculator to simplify expressions in scientific notation.