# 1.2 Exponents and scientific notation  (Page 6/9)

 Page 6 / 9

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!

#### Questions & Answers

How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

 By By By By Rhodes By Richley Crapo By OpenStax By Robert Murphy By Maureen Miller By Sarah Warren By Richley Crapo By Edgar Delgado By OpenStax