1.2 Exponents and scientific notation  (Page 6/9)

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Be careful not to include the leading 0 in your count. We move the decimal point 13 places to the right, so the exponent of 10 is 13. The exponent is negative because we moved the decimal point to the right. This is what we should expect for a small number.

$4.7\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{-13}$

Scientific notation

A number is written in scientific notation    if it is written in the form $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{n},$ where $\text{\hspace{0.17em}}1\le |a|<10\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is an integer.

Converting standard notation to scientific notation

Write each number in scientific notation.

1. Distance to Andromeda Galaxy from Earth: 24,000,000,000,000,000,000,000 m
2. Diameter of Andromeda Galaxy: 1,300,000,000,000,000,000,000 m
3. Number of stars in Andromeda Galaxy: 1,000,000,000,000
4. Diameter of electron: 0.00000000000094 m
5. Probability of being struck by lightning in any single year: 0.00000143

Write each number in scientific notation.

1. U.S. national debt per taxpayer (April 2014): $152,000 2. World population (April 2014): 7,158,000,000 3. World gross national income (April 2014):$85,500,000,000,000
4. Time for light to travel 1 m: 0.00000000334 s
5. Probability of winning lottery (match 6 of 49 possible numbers): 0.0000000715
1. $1.52×{10}^{5}$
2. $7.158×{10}^{9}$
3. $8.55×{10}^{13}$
4. $3.34×{10}^{-9}$
5. $7.15×{10}^{-8}$

Converting from scientific to standard notation

To convert a number in scientific notation    to standard notation, simply reverse the process. Move the decimal $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ places to the right if $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is positive or $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ places to the left if $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is negative and add zeros as needed. Remember, if $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is positive, the value of the number is greater than 1, and if $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is negative, the value of the number is less than one.

Converting scientific notation to standard notation

Convert each number in scientific notation to standard notation.

1. $3.547\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{14}$
2. $-2\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{6}$
3. $7.91\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{-7}$
4. $-8.05\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{-12}$

Convert each number in scientific notation to standard notation.

1. $7.03\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{5}$
2. $-8.16\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{11}$
3. $-3.9\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{-13}$
4. $8\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{-6}$
1. $703,000$
2. $-816,000,000,000$
3. $-0.000\text{\hspace{0.17em}}000\text{\hspace{0.17em}}000\text{\hspace{0.17em}}000\text{\hspace{0.17em}}39$
4. $0.000008$

Using scientific notation in applications

Scientific notation, used with the rules of exponents, makes calculating with large or small numbers much easier than doing so using standard notation. For example, suppose we are asked to calculate the number of atoms in 1 L of water. Each water molecule contains 3 atoms (2 hydrogen and 1 oxygen). The average drop of water contains around $\text{\hspace{0.17em}}1.32\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{21}\text{\hspace{0.17em}}$ molecules of water and 1 L of water holds about $\text{\hspace{0.17em}}1.22\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{4}\text{\hspace{0.17em}}$ average drops. Therefore, there are approximately $\text{\hspace{0.17em}}3\cdot \left(1.32\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{21}\right)\cdot \left(1.22\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{4}\right)\approx 4.83\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{10}^{25}\text{\hspace{0.17em}}$ atoms in 1 L of water. We simply multiply the decimal terms and add the exponents. Imagine having to perform the calculation without using scientific notation!

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