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# Scientific Notation with a Calculator

## Use technology to perform operations on numbers in scientific notation

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Scientific Notation with a Calculator

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.

### Scientific Notation with a Calculator

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

#### Let's evaluate 2.6×105\begin{align*}2.6 \times 10^5\end{align*} using a scientific calculator:

To enter 2.6×105\begin{align*}2.6 \times 10^5\end{align*}, enter 2.6 [EE] 5.

When you hit [ENTER] the calculator displays 2.6E5\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.

#### Now, let's simplify (3.2×106)⋅(8.7×1011)\begin{align*}(3.2 \times 10^6) \cdot (8.7 \times 10^{11})\end{align*}:

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

(3.2×106)(8.7×1011)=3.2×8.7106×1011=27.84×1017=2.784×101×1017=2.784×1018\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 27.84=2.784×101\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.

#### Finally, let's use scientific notation to solve the following problem:

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.

We know that percent means we divide by 100, and so our calculation for the mass (in kg) is: 1100×11,000,000×11,000,000,000×11,000,000,000=102×106×109×109\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.

102×106×109×109=10((2)+(6)+(9)+(9))=1026 kg.\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 1×1026 kg\begin{align*}1 \times 10^{-26} \ kg\end{align*}.

### Examples

#### Example 1

Earlier, you were told that there are about 7 billion people living on the Earth and that each person is made up of about 100 trillion cells. About how many cells make up all the people on earth?

We want to multiply 7,000,000,000 by 100,000,000,000,000. You can either do this with your calculator or by hand.

In scientific notation, we want to multiply (7×109)(1×1014)\begin{align*}(7\times 10^9) \cdot (1\times 10^{14})\end{align*}. By hand we get:

(7×109)(1×1014)=7×1×109×1014=7×1023\begin{align*}(7\times 10^9) \cdot (1\times 10^{14})&=7\times 1 \times 10^9\times 10^{14}\\ &=7\times 10^{23}\end{align*}

To calculate this with your calculator, type in 7 [EE] 9 and multiply that by 1 [EE] 14. You should get 7 E 23 as your result.

#### Example 2

(1.7×106)(2.7×1011)\begin{align*}(1.7 \times 10^6) \cdot (2.7 \times 10^{-11})\end{align*}

(1.7×106)(2.7×1011)=1.7×2.7106×1011=4.59×105\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*}

#### Example 3

(1.03×108)÷(3.25×105)\begin{align*}(1.03 \times 10^8) \div (-3.25 \times 10^{-5})\end{align*}

(1.03×108)÷(3.25×105)=0.317×108(5)=1.03×1083.25×105=1.033.25×108105=0.317×1013=3.17×1012\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.

### Review

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

1. (3.2×106)(8.7×1011)\begin{align*}(3.2 \times 10^6) \cdot (8.7 \times 10^{11})\end{align*}
2. (5.2×104)(3.8×1019)\begin{align*}(5.2 \times 10^{-4}) \cdot (3.8 \times 10^{-19})\end{align*}
3. (1.7×106)(2.7×1011)\begin{align*}(1.7 \times 10^6) \cdot (2.7 \times 10^{-11})\end{align*}
4. (3.2×106)÷(8.7×1011)\begin{align*}(3.2 \times 10^6) \div (8.7 \times 10^{11})\end{align*}
5. (5.2×104)÷(3.8×1019)\begin{align*}(5.2 \times 10^{-4}) \div (3.8 \times 10^{-19})\end{align*}
6. (1.7×106)÷(2.7×1011)\begin{align*}(1.7 \times 10^6) \div (2.7 \times 10^{-11})\end{align*}
7. (4.7×103)(2.35×1027)\begin{align*}(4.7 \times 10^3) \cdot (2.35 \times 10^{-27})\end{align*}
8. (1.05×1016)(7.003×1021)\begin{align*}(1.05 \times 10^{-16}) \cdot (7.003 \times 10^{21})\end{align*}
9. (2.09×105)÷(5.006×103)\begin{align*}(2.09 \times 10^5) \div (5.006 \times 10^{-3})\end{align*}
10. (9.01×1022)÷(2.6×1033)\begin{align*}(9.01 \times 10^{22}) \div (2.6 \times 10^{33})\end{align*}

Mixed Review

1. 14 milliliters of a 40% sugar solution was mixed with 4 milliliters of pure water. What is the concentration of the mixture?
2. Solve the system \begin{align*}\begin{cases} 6x+3y=18\\ \text{-}15=11y-5x \end{cases}\end{align*}.
3. 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*}.
4. Simplify \begin{align*}\frac{5a^6 b^2 c^{-6}}{a^{11} b}\end{align*}. Your answer should have only positive exponents.
5. 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?
6. The volume of a 3-dimensional 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?

To see the Review answers, open this PDF file and look for section 8.6.

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### Vocabulary Language: English Spanish

scientific notation

A number is expressed in scientific notation when it is in the form $N \times 10^n$ where $1\le N <10$ and $n$ is an integer.